Prompt:
<|im_start|>user
Preamble: Sebastian owns a coffee factory in Argentina. His production function is:
\[
F(K, L)=(K-1)^{\frac{1}{4}} L^{\frac{1}{4}}
\]
Consider the cost of capital to be $r$ and the wage to be $w$. Both inputs are variable, and Sebastian faces no fixed costs.

What is the marginal rate of technical substitution of labor for capital?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Write the condition which involves the SRAC, or short-run average cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Write the condition which involves the SRAC, or short-run average cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Write the condition which involves the SRAC, or short-run average cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Write the condition which involves the SRAC, or short-run average cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Subproblem 0: Write the condition which involves the SRAC, or short-run average cost?

Solution: \boxed{SRAC=LRAC}, short-run average cost equals long-run average cost.

Final answer: The final answer is SRAC=LRAC. I hope it is correct.

Subproblem 1: Write the condition which involves SRMC, or short-run marginal cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Subproblem 0: Write the condition which involves the SRAC, or short-run average cost?

Solution: \boxed{SRAC=LRAC}, short-run average cost equals long-run average cost.

Final answer: The final answer is SRAC=LRAC. I hope it is correct.

Subproblem 1: Write the condition which involves SRMC, or short-run marginal cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Subproblem 0: Write the condition which involves the SRAC, or short-run average cost?

Solution: \boxed{SRAC=LRAC}, short-run average cost equals long-run average cost.

Final answer: The final answer is SRAC=LRAC. I hope it is correct.

Subproblem 1: Write the condition which involves SRMC, or short-run marginal cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: There are two algebraic conditions describing a firm that is at a capital level that minimizes its costs in the long-term.

Subproblem 0: Write the condition which involves the SRAC, or short-run average cost?

Solution: \boxed{SRAC=LRAC}, short-run average cost equals long-run average cost.

Final answer: The final answer is SRAC=LRAC. I hope it is correct.

Subproblem 1: Write the condition which involves SRMC, or short-run marginal cost?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Obtain the short run industry supply function for widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Obtain the short run industry supply function for widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Obtain the short run industry supply function for widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Obtain the short run industry supply function for widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Moldavia is a small country that currently trades freely in the world barley market. Demand and supply for barley in Moldavia is governed by the following schedules:
Demand: $Q^{D}=4-P$
Supply: $Q^{S}=P$
The world price of barley is $\$ 1 /$ bushel.

Calculate the free trade equilibrium price of barley in Moldavia, in dollars per bushel.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Moldavia is a small country that currently trades freely in the world barley market. Demand and supply for barley in Moldavia is governed by the following schedules:
Demand: $Q^{D}=4-P$
Supply: $Q^{S}=P$
The world price of barley is $\$ 1 /$ bushel.

Calculate the free trade equilibrium price of barley in Moldavia, in dollars per bushel.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Moldavia is a small country that currently trades freely in the world barley market. Demand and supply for barley in Moldavia is governed by the following schedules:
Demand: $Q^{D}=4-P$
Supply: $Q^{S}=P$
The world price of barley is $\$ 1 /$ bushel.

Calculate the free trade equilibrium price of barley in Moldavia, in dollars per bushel.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Moldavia is a small country that currently trades freely in the world barley market. Demand and supply for barley in Moldavia is governed by the following schedules:
Demand: $Q^{D}=4-P$
Supply: $Q^{S}=P$
The world price of barley is $\$ 1 /$ bushel.

Calculate the free trade equilibrium price of barley in Moldavia, in dollars per bushel.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Subproblem 0: Obtain the short run industry supply function for widgets.

Solution: Since $P=M C=y^{2}$, the supply function of each firm is given by $y_{i}^{s}=p^{\frac{1}{2}}$.
The industry supply function is $y^{s}(p)=100 y_{i}^{s}(p)=\boxed{100 p^{\frac{1}{2}}}$.

Final answer: The final answer is 100 p^{\frac{1}{2}}. I hope it is correct.

Subproblem 1: Obtain the short run equilibrium price of widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Subproblem 0: Obtain the short run industry supply function for widgets.

Solution: Since $P=M C=y^{2}$, the supply function of each firm is given by $y_{i}^{s}=p^{\frac{1}{2}}$.
The industry supply function is $y^{s}(p)=100 y_{i}^{s}(p)=\boxed{100 p^{\frac{1}{2}}}$.

Final answer: The final answer is 100 p^{\frac{1}{2}}. I hope it is correct.

Subproblem 1: Obtain the short run equilibrium price of widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Subproblem 0: Obtain the short run industry supply function for widgets.

Solution: Since $P=M C=y^{2}$, the supply function of each firm is given by $y_{i}^{s}=p^{\frac{1}{2}}$.
The industry supply function is $y^{s}(p)=100 y_{i}^{s}(p)=\boxed{100 p^{\frac{1}{2}}}$.

Final answer: The final answer is 100 p^{\frac{1}{2}}. I hope it is correct.

Subproblem 1: Obtain the short run equilibrium price of widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose, in the short run, the output of widgets is supplied by 100 identical competitive firms, each having a cost function:
\[
c_{s}(y)=\frac{1}{3} y^{3}+2
\]
The demand for widgets is given by:
\[
y^{d}(p)=6400 / p^{\frac{1}{2}}
\]

Subproblem 0: Obtain the short run industry supply function for widgets.

Solution: Since $P=M C=y^{2}$, the supply function of each firm is given by $y_{i}^{s}=p^{\frac{1}{2}}$.
The industry supply function is $y^{s}(p)=100 y_{i}^{s}(p)=\boxed{100 p^{\frac{1}{2}}}$.

Final answer: The final answer is 100 p^{\frac{1}{2}}. I hope it is correct.

Subproblem 1: Obtain the short run equilibrium price of widgets.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A consumer's preferences are representable by the following utility function:
\[
  u(x, y)=x^{\frac{1}{2}}+y
\]

Subproblem 0: Obtain the marginal rate of substitution of the consumer at an arbitrary point $(X,Y)$, where $X>0$ and $Y>0$.

Solution: \[ M R S=-\frac{\frac{1}{2} x^{-\frac{1}{2}}}{1}=\boxed{-\frac{1}{2} X^{-\frac{1}{2}}} \]

Final answer: The final answer is -\frac{1}{2} X^{-\frac{1}{2}}. I hope it is correct.

Subproblem 1: Suppose the price of the second good $(y)$ is 1 , and the price of the first good $(x)$ is denoted by $p>0$. If the consumer's income is $m>\frac{1}{4p}$, in the optimal consumption bundle of the consumer (in terms of $m$ and $p$ ), what is the quantity of the first good $(x)$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A consumer's preferences are representable by the following utility function:
\[
  u(x, y)=x^{\frac{1}{2}}+y
\]

Subproblem 0: Obtain the marginal rate of substitution of the consumer at an arbitrary point $(X,Y)$, where $X>0$ and $Y>0$.

Solution: \[ M R S=-\frac{\frac{1}{2} x^{-\frac{1}{2}}}{1}=\boxed{-\frac{1}{2} X^{-\frac{1}{2}}} \]

Final answer: The final answer is -\frac{1}{2} X^{-\frac{1}{2}}. I hope it is correct.

Subproblem 1: Suppose the price of the second good $(y)$ is 1 , and the price of the first good $(x)$ is denoted by $p>0$. If the consumer's income is $m>\frac{1}{4p}$, in the optimal consumption bundle of the consumer (in terms of $m$ and $p$ ), what is the quantity of the first good $(x)$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A consumer's preferences are representable by the following utility function:
\[
  u(x, y)=x^{\frac{1}{2}}+y
\]

Subproblem 0: Obtain the marginal rate of substitution of the consumer at an arbitrary point $(X,Y)$, where $X>0$ and $Y>0$.

Solution: \[ M R S=-\frac{\frac{1}{2} x^{-\frac{1}{2}}}{1}=\boxed{-\frac{1}{2} X^{-\frac{1}{2}}} \]

Final answer: The final answer is -\frac{1}{2} X^{-\frac{1}{2}}. I hope it is correct.

Subproblem 1: Suppose the price of the second good $(y)$ is 1 , and the price of the first good $(x)$ is denoted by $p>0$. If the consumer's income is $m>\frac{1}{4p}$, in the optimal consumption bundle of the consumer (in terms of $m$ and $p$ ), what is the quantity of the first good $(x)$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A consumer's preferences are representable by the following utility function:
\[
  u(x, y)=x^{\frac{1}{2}}+y
\]

Subproblem 0: Obtain the marginal rate of substitution of the consumer at an arbitrary point $(X,Y)$, where $X>0$ and $Y>0$.

Solution: \[ M R S=-\frac{\frac{1}{2} x^{-\frac{1}{2}}}{1}=\boxed{-\frac{1}{2} X^{-\frac{1}{2}}} \]

Final answer: The final answer is -\frac{1}{2} X^{-\frac{1}{2}}. I hope it is correct.

Subproblem 1: Suppose the price of the second good $(y)$ is 1 , and the price of the first good $(x)$ is denoted by $p>0$. If the consumer's income is $m>\frac{1}{4p}$, in the optimal consumption bundle of the consumer (in terms of $m$ and $p$ ), what is the quantity of the first good $(x)$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the market for apple juice. In this market, the supply curve is given by $Q_{S}=$ $10 P_{J}-5 P_{A}$ and the demand curve is given by $Q_{D}=100-15 P_{J}+10 P_{T}$, where $J$ denotes apple juice, $A$ denotes apples, and $T$ denotes tea.

Assume that $P_{A}$ is fixed at $\$ 1$ and $P_{T}=5$. Calculate the equilibrium price in the apple juice market.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the market for apple juice. In this market, the supply curve is given by $Q_{S}=$ $10 P_{J}-5 P_{A}$ and the demand curve is given by $Q_{D}=100-15 P_{J}+10 P_{T}$, where $J$ denotes apple juice, $A$ denotes apples, and $T$ denotes tea.

Assume that $P_{A}$ is fixed at $\$ 1$ and $P_{T}=5$. Calculate the equilibrium price in the apple juice market.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the market for apple juice. In this market, the supply curve is given by $Q_{S}=$ $10 P_{J}-5 P_{A}$ and the demand curve is given by $Q_{D}=100-15 P_{J}+10 P_{T}$, where $J$ denotes apple juice, $A$ denotes apples, and $T$ denotes tea.

Assume that $P_{A}$ is fixed at $\$ 1$ and $P_{T}=5$. Calculate the equilibrium price in the apple juice market.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the market for apple juice. In this market, the supply curve is given by $Q_{S}=$ $10 P_{J}-5 P_{A}$ and the demand curve is given by $Q_{D}=100-15 P_{J}+10 P_{T}$, where $J$ denotes apple juice, $A$ denotes apples, and $T$ denotes tea.

Assume that $P_{A}$ is fixed at $\$ 1$ and $P_{T}=5$. Calculate the equilibrium price in the apple juice market.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In Cambridge, shoppers can buy apples from two sources: a local orchard, and a store that ships apples from out of state. The orchard can produce up to 50 apples per day at a constant marginal cost of 25 cents per apple. The store can supply any remaining apples demanded, at a constant marginal cost of 75 cents per unit. When apples cost 75 cents per apple, the residents of Cambridge buy 150 apples in a day.

Assume that the city of Cambridge sets the price of apples within its borders. What price should it set, in cents?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In Cambridge, shoppers can buy apples from two sources: a local orchard, and a store that ships apples from out of state. The orchard can produce up to 50 apples per day at a constant marginal cost of 25 cents per apple. The store can supply any remaining apples demanded, at a constant marginal cost of 75 cents per unit. When apples cost 75 cents per apple, the residents of Cambridge buy 150 apples in a day.

Assume that the city of Cambridge sets the price of apples within its borders. What price should it set, in cents?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In Cambridge, shoppers can buy apples from two sources: a local orchard, and a store that ships apples from out of state. The orchard can produce up to 50 apples per day at a constant marginal cost of 25 cents per apple. The store can supply any remaining apples demanded, at a constant marginal cost of 75 cents per unit. When apples cost 75 cents per apple, the residents of Cambridge buy 150 apples in a day.

Assume that the city of Cambridge sets the price of apples within its borders. What price should it set, in cents?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: In Cambridge, shoppers can buy apples from two sources: a local orchard, and a store that ships apples from out of state. The orchard can produce up to 50 apples per day at a constant marginal cost of 25 cents per apple. The store can supply any remaining apples demanded, at a constant marginal cost of 75 cents per unit. When apples cost 75 cents per apple, the residents of Cambridge buy 150 apples in a day.

Assume that the city of Cambridge sets the price of apples within its borders. What price should it set, in cents?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You manage a factory that produces cans of peanut butter. The current market price is $\$ 10 /$ can, and you know the following about your costs (MC stands for marginal cost, and ATC stands for average total cost):
\[
\begin{array}{l}
MC(5)=10 \\
ATC(5)=6 \\
MC(4)=4 \\
ATC(4)=4
\end{array}
\]

A case of food poisoning breaks out due to your peanut butter, and you lose a lawsuit against your company. As punishment, Judge Judy decides to take away all of your profits, and considers the following two options to be equivalent:
i. Pay a lump sum in the amount of your profits.
ii. Impose a tax of $\$\left[P-A T C\left(q^{*}\right)\right]$ per can since that is your current profit per can, where $q^{*}$ is the profit maximizing output before the lawsuit.
How much is the tax, in dollars per can?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You manage a factory that produces cans of peanut butter. The current market price is $\$ 10 /$ can, and you know the following about your costs (MC stands for marginal cost, and ATC stands for average total cost):
\[
\begin{array}{l}
MC(5)=10 \\
ATC(5)=6 \\
MC(4)=4 \\
ATC(4)=4
\end{array}
\]

A case of food poisoning breaks out due to your peanut butter, and you lose a lawsuit against your company. As punishment, Judge Judy decides to take away all of your profits, and considers the following two options to be equivalent:
i. Pay a lump sum in the amount of your profits.
ii. Impose a tax of $\$\left[P-A T C\left(q^{*}\right)\right]$ per can since that is your current profit per can, where $q^{*}$ is the profit maximizing output before the lawsuit.
How much is the tax, in dollars per can?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You manage a factory that produces cans of peanut butter. The current market price is $\$ 10 /$ can, and you know the following about your costs (MC stands for marginal cost, and ATC stands for average total cost):
\[
\begin{array}{l}
MC(5)=10 \\
ATC(5)=6 \\
MC(4)=4 \\
ATC(4)=4
\end{array}
\]

A case of food poisoning breaks out due to your peanut butter, and you lose a lawsuit against your company. As punishment, Judge Judy decides to take away all of your profits, and considers the following two options to be equivalent:
i. Pay a lump sum in the amount of your profits.
ii. Impose a tax of $\$\left[P-A T C\left(q^{*}\right)\right]$ per can since that is your current profit per can, where $q^{*}$ is the profit maximizing output before the lawsuit.
How much is the tax, in dollars per can?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You manage a factory that produces cans of peanut butter. The current market price is $\$ 10 /$ can, and you know the following about your costs (MC stands for marginal cost, and ATC stands for average total cost):
\[
\begin{array}{l}
MC(5)=10 \\
ATC(5)=6 \\
MC(4)=4 \\
ATC(4)=4
\end{array}
\]

A case of food poisoning breaks out due to your peanut butter, and you lose a lawsuit against your company. As punishment, Judge Judy decides to take away all of your profits, and considers the following two options to be equivalent:
i. Pay a lump sum in the amount of your profits.
ii. Impose a tax of $\$\left[P-A T C\left(q^{*}\right)\right]$ per can since that is your current profit per can, where $q^{*}$ is the profit maximizing output before the lawsuit.
How much is the tax, in dollars per can?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose there are exactly two consumers (Albie and Bubbie) who demand strawberries. Suppose that Albie's demand for strawberries is given by
\[
q_{a}(p)=p^{\alpha} f_{a}\left(I_{a}\right)
\]
and Bubbie's demand is given by
\[
q_{b}(p)=p^{\beta} f_{b}\left(I_{b}\right)
\]
where $I_{a}$ and $I_{b}$ are Albie and Bubbie's incomes, and $f_{a}(\cdot)$ and $f_{b}(\cdot)$ are two unknown functions.

Find Albie's (own-price) elasticity of demand, $\epsilon_{q_{a}, p}$. Use the sign convention that $\epsilon_{y, x}=\frac{\partial y}{\partial x} \frac{x}{y}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose there are exactly two consumers (Albie and Bubbie) who demand strawberries. Suppose that Albie's demand for strawberries is given by
\[
q_{a}(p)=p^{\alpha} f_{a}\left(I_{a}\right)
\]
and Bubbie's demand is given by
\[
q_{b}(p)=p^{\beta} f_{b}\left(I_{b}\right)
\]
where $I_{a}$ and $I_{b}$ are Albie and Bubbie's incomes, and $f_{a}(\cdot)$ and $f_{b}(\cdot)$ are two unknown functions.

Find Albie's (own-price) elasticity of demand, $\epsilon_{q_{a}, p}$. Use the sign convention that $\epsilon_{y, x}=\frac{\partial y}{\partial x} \frac{x}{y}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose there are exactly two consumers (Albie and Bubbie) who demand strawberries. Suppose that Albie's demand for strawberries is given by
\[
q_{a}(p)=p^{\alpha} f_{a}\left(I_{a}\right)
\]
and Bubbie's demand is given by
\[
q_{b}(p)=p^{\beta} f_{b}\left(I_{b}\right)
\]
where $I_{a}$ and $I_{b}$ are Albie and Bubbie's incomes, and $f_{a}(\cdot)$ and $f_{b}(\cdot)$ are two unknown functions.

Find Albie's (own-price) elasticity of demand, $\epsilon_{q_{a}, p}$. Use the sign convention that $\epsilon_{y, x}=\frac{\partial y}{\partial x} \frac{x}{y}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Suppose there are exactly two consumers (Albie and Bubbie) who demand strawberries. Suppose that Albie's demand for strawberries is given by
\[
q_{a}(p)=p^{\alpha} f_{a}\left(I_{a}\right)
\]
and Bubbie's demand is given by
\[
q_{b}(p)=p^{\beta} f_{b}\left(I_{b}\right)
\]
where $I_{a}$ and $I_{b}$ are Albie and Bubbie's incomes, and $f_{a}(\cdot)$ and $f_{b}(\cdot)$ are two unknown functions.

Find Albie's (own-price) elasticity of demand, $\epsilon_{q_{a}, p}$. Use the sign convention that $\epsilon_{y, x}=\frac{\partial y}{\partial x} \frac{x}{y}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You have been asked to analyze the market for steel. From public sources, you are able to find that last year's price for steel was $\$ 20$ per ton. At this price, 100 million tons were sold on the world market. From trade association data you are able to obtain estimates for the own price elasticities of demand and supply on the world markets as $-0.25$ for demand and $0.5$ for supply. Assume that steel has linear demand and supply curves throughout, and that the market is competitive.

Solve for the equations of demand in this market.  Use $P$ to represent the price of steel in dollars per ton, and $X_{d}$ to represent the demand in units of millions of tons.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You have been asked to analyze the market for steel. From public sources, you are able to find that last year's price for steel was $\$ 20$ per ton. At this price, 100 million tons were sold on the world market. From trade association data you are able to obtain estimates for the own price elasticities of demand and supply on the world markets as $-0.25$ for demand and $0.5$ for supply. Assume that steel has linear demand and supply curves throughout, and that the market is competitive.

Solve for the equations of demand in this market.  Use $P$ to represent the price of steel in dollars per ton, and $X_{d}$ to represent the demand in units of millions of tons.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You have been asked to analyze the market for steel. From public sources, you are able to find that last year's price for steel was $\$ 20$ per ton. At this price, 100 million tons were sold on the world market. From trade association data you are able to obtain estimates for the own price elasticities of demand and supply on the world markets as $-0.25$ for demand and $0.5$ for supply. Assume that steel has linear demand and supply curves throughout, and that the market is competitive.

Solve for the equations of demand in this market.  Use $P$ to represent the price of steel in dollars per ton, and $X_{d}$ to represent the demand in units of millions of tons.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: You have been asked to analyze the market for steel. From public sources, you are able to find that last year's price for steel was $\$ 20$ per ton. At this price, 100 million tons were sold on the world market. From trade association data you are able to obtain estimates for the own price elasticities of demand and supply on the world markets as $-0.25$ for demand and $0.5$ for supply. Assume that steel has linear demand and supply curves throughout, and that the market is competitive.

Solve for the equations of demand in this market.  Use $P$ to represent the price of steel in dollars per ton, and $X_{d}$ to represent the demand in units of millions of tons.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Harmonic Oscillator Subjected to Perturbation by an Electric Field: An electron is connected by a harmonic spring to a fixed point at $x=0$. It is subject to a field-free potential energy
\[
V(x)=\frac{1}{2} k x^{2} .
\]
The energy levels and eigenstates are those of a harmonic oscillator where
\[
\begin{aligned}
\omega &=\left[k / m_{e}\right]^{1 / 2} \\
E_{v} &=\hbar \omega(v+1 / 2) \\
\psi_{v}(x) &=(v !)^{-1 / 2}\left(\hat{\boldsymbol{a}}^{\dagger}\right)^{v} \psi_{v=0}(x) .
\end{aligned}
\]
Now a constant electric field, $E_{0}$, is applied and $V(x)$ becomes
\[
V(x)=\frac{1}{2} k x^{2}+E_{0} e x \quad(e>0 \text { by definition }) .
\]
Write an expression for the energy levels $E_{v}$ as a function of the strength of the electric field.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Harmonic Oscillator Subjected to Perturbation by an Electric Field: An electron is connected by a harmonic spring to a fixed point at $x=0$. It is subject to a field-free potential energy
\[
V(x)=\frac{1}{2} k x^{2} .
\]
The energy levels and eigenstates are those of a harmonic oscillator where
\[
\begin{aligned}
\omega &=\left[k / m_{e}\right]^{1 / 2} \\
E_{v} &=\hbar \omega(v+1 / 2) \\
\psi_{v}(x) &=(v !)^{-1 / 2}\left(\hat{\boldsymbol{a}}^{\dagger}\right)^{v} \psi_{v=0}(x) .
\end{aligned}
\]
Now a constant electric field, $E_{0}$, is applied and $V(x)$ becomes
\[
V(x)=\frac{1}{2} k x^{2}+E_{0} e x \quad(e>0 \text { by definition }) .
\]
Write an expression for the energy levels $E_{v}$ as a function of the strength of the electric field.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Harmonic Oscillator Subjected to Perturbation by an Electric Field: An electron is connected by a harmonic spring to a fixed point at $x=0$. It is subject to a field-free potential energy
\[
V(x)=\frac{1}{2} k x^{2} .
\]
The energy levels and eigenstates are those of a harmonic oscillator where
\[
\begin{aligned}
\omega &=\left[k / m_{e}\right]^{1 / 2} \\
E_{v} &=\hbar \omega(v+1 / 2) \\
\psi_{v}(x) &=(v !)^{-1 / 2}\left(\hat{\boldsymbol{a}}^{\dagger}\right)^{v} \psi_{v=0}(x) .
\end{aligned}
\]
Now a constant electric field, $E_{0}$, is applied and $V(x)$ becomes
\[
V(x)=\frac{1}{2} k x^{2}+E_{0} e x \quad(e>0 \text { by definition }) .
\]
Write an expression for the energy levels $E_{v}$ as a function of the strength of the electric field.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Harmonic Oscillator Subjected to Perturbation by an Electric Field: An electron is connected by a harmonic spring to a fixed point at $x=0$. It is subject to a field-free potential energy
\[
V(x)=\frac{1}{2} k x^{2} .
\]
The energy levels and eigenstates are those of a harmonic oscillator where
\[
\begin{aligned}
\omega &=\left[k / m_{e}\right]^{1 / 2} \\
E_{v} &=\hbar \omega(v+1 / 2) \\
\psi_{v}(x) &=(v !)^{-1 / 2}\left(\hat{\boldsymbol{a}}^{\dagger}\right)^{v} \psi_{v=0}(x) .
\end{aligned}
\]
Now a constant electric field, $E_{0}$, is applied and $V(x)$ becomes
\[
V(x)=\frac{1}{2} k x^{2}+E_{0} e x \quad(e>0 \text { by definition }) .
\]
Write an expression for the energy levels $E_{v}$ as a function of the strength of the electric field.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: The following concern the independent particle model. You may find the following set of Coulomb and exchange integrals useful (energies in $\mathrm{eV}$):
$\mathrm{J}_{1 s 1 s}=17.0 Z$
$\mathrm{~J}_{1 s 2 s}=4.8 Z$
$\mathrm{~K}_{1 s 2 s}=0.9 Z$
$\mathrm{~J}_{2 s 2 s}=3.5 Z$
$\mathrm{J}_{1 s 2 p}=6.6 Z$
$\mathrm{~K}_{1 s 2 p}=0.5 Z$
$\mathrm{~J}_{2 s 2 p}=4.4 Z$
$\mathrm{~K}_{2 s 2 p}=0.8 Z$
$\mathrm{J}_{2 p_{i}, 2 p_{i}}=3.9 Z$
$\mathrm{~J}_{2 p_{i}, 2 p_{k}}=3.5 Z$
$\mathrm{~K}_{2 p_{i}, 2 p_{k}}=0.2 Z i \neq k$

Using the independent particle model, what is the energy difference between the $1 s^{2} 2 p_{x}^{2}$ configuration and the $1 s^{2} 2 s^{2}$ configuration? Give your answer in eV, in terms of $Z$, and round to a single decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: The following concern the independent particle model. You may find the following set of Coulomb and exchange integrals useful (energies in $\mathrm{eV}$):
$\mathrm{J}_{1 s 1 s}=17.0 Z$
$\mathrm{~J}_{1 s 2 s}=4.8 Z$
$\mathrm{~K}_{1 s 2 s}=0.9 Z$
$\mathrm{~J}_{2 s 2 s}=3.5 Z$
$\mathrm{J}_{1 s 2 p}=6.6 Z$
$\mathrm{~K}_{1 s 2 p}=0.5 Z$
$\mathrm{~J}_{2 s 2 p}=4.4 Z$
$\mathrm{~K}_{2 s 2 p}=0.8 Z$
$\mathrm{J}_{2 p_{i}, 2 p_{i}}=3.9 Z$
$\mathrm{~J}_{2 p_{i}, 2 p_{k}}=3.5 Z$
$\mathrm{~K}_{2 p_{i}, 2 p_{k}}=0.2 Z i \neq k$

Using the independent particle model, what is the energy difference between the $1 s^{2} 2 p_{x}^{2}$ configuration and the $1 s^{2} 2 s^{2}$ configuration? Give your answer in eV, in terms of $Z$, and round to a single decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: The following concern the independent particle model. You may find the following set of Coulomb and exchange integrals useful (energies in $\mathrm{eV}$):
$\mathrm{J}_{1 s 1 s}=17.0 Z$
$\mathrm{~J}_{1 s 2 s}=4.8 Z$
$\mathrm{~K}_{1 s 2 s}=0.9 Z$
$\mathrm{~J}_{2 s 2 s}=3.5 Z$
$\mathrm{J}_{1 s 2 p}=6.6 Z$
$\mathrm{~K}_{1 s 2 p}=0.5 Z$
$\mathrm{~J}_{2 s 2 p}=4.4 Z$
$\mathrm{~K}_{2 s 2 p}=0.8 Z$
$\mathrm{J}_{2 p_{i}, 2 p_{i}}=3.9 Z$
$\mathrm{~J}_{2 p_{i}, 2 p_{k}}=3.5 Z$
$\mathrm{~K}_{2 p_{i}, 2 p_{k}}=0.2 Z i \neq k$

Using the independent particle model, what is the energy difference between the $1 s^{2} 2 p_{x}^{2}$ configuration and the $1 s^{2} 2 s^{2}$ configuration? Give your answer in eV, in terms of $Z$, and round to a single decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: The following concern the independent particle model. You may find the following set of Coulomb and exchange integrals useful (energies in $\mathrm{eV}$):
$\mathrm{J}_{1 s 1 s}=17.0 Z$
$\mathrm{~J}_{1 s 2 s}=4.8 Z$
$\mathrm{~K}_{1 s 2 s}=0.9 Z$
$\mathrm{~J}_{2 s 2 s}=3.5 Z$
$\mathrm{J}_{1 s 2 p}=6.6 Z$
$\mathrm{~K}_{1 s 2 p}=0.5 Z$
$\mathrm{~J}_{2 s 2 p}=4.4 Z$
$\mathrm{~K}_{2 s 2 p}=0.8 Z$
$\mathrm{J}_{2 p_{i}, 2 p_{i}}=3.9 Z$
$\mathrm{~J}_{2 p_{i}, 2 p_{k}}=3.5 Z$
$\mathrm{~K}_{2 p_{i}, 2 p_{k}}=0.2 Z i \neq k$

Using the independent particle model, what is the energy difference between the $1 s^{2} 2 p_{x}^{2}$ configuration and the $1 s^{2} 2 s^{2}$ configuration? Give your answer in eV, in terms of $Z$, and round to a single decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A pulsed Nd:YAG laser is found in many physical chemistry laboratories.

For a $2.00 \mathrm{~mJ}$ pulse of laser light, how many photons are there at $1.06 \mu \mathrm{m}$ (the Nd:YAG fundamental) in the pulse?  PAnswer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A pulsed Nd:YAG laser is found in many physical chemistry laboratories.

For a $2.00 \mathrm{~mJ}$ pulse of laser light, how many photons are there at $1.06 \mu \mathrm{m}$ (the Nd:YAG fundamental) in the pulse?  PAnswer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A pulsed Nd:YAG laser is found in many physical chemistry laboratories.

For a $2.00 \mathrm{~mJ}$ pulse of laser light, how many photons are there at $1.06 \mu \mathrm{m}$ (the Nd:YAG fundamental) in the pulse?  PAnswer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: A pulsed Nd:YAG laser is found in many physical chemistry laboratories.

For a $2.00 \mathrm{~mJ}$ pulse of laser light, how many photons are there at $1.06 \mu \mathrm{m}$ (the Nd:YAG fundamental) in the pulse?  PAnswer to three significant figures.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Given that the work function of chromium is $4.40 \mathrm{eV}$, calculate the kinetic energy of electrons in Joules emitted from a clean chromium surface that is irradiated with ultraviolet radiation of wavelength $200 \mathrm{~nm}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Given that the work function of chromium is $4.40 \mathrm{eV}$, calculate the kinetic energy of electrons in Joules emitted from a clean chromium surface that is irradiated with ultraviolet radiation of wavelength $200 \mathrm{~nm}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Given that the work function of chromium is $4.40 \mathrm{eV}$, calculate the kinetic energy of electrons in Joules emitted from a clean chromium surface that is irradiated with ultraviolet radiation of wavelength $200 \mathrm{~nm}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Given that the work function of chromium is $4.40 \mathrm{eV}$, calculate the kinetic energy of electrons in Joules emitted from a clean chromium surface that is irradiated with ultraviolet radiation of wavelength $200 \mathrm{~nm}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Compute the momentum of one $500 \mathrm{~nm}$ photon using $p_{\text {photon }}=E_{\text {photon }} / c$ where $c$ is the speed of light, $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$, and $\nu=c / \lambda$.  Express your answer in kilogram meters per second, rounding your answer to three decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Compute the momentum of one $500 \mathrm{~nm}$ photon using $p_{\text {photon }}=E_{\text {photon }} / c$ where $c$ is the speed of light, $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$, and $\nu=c / \lambda$.  Express your answer in kilogram meters per second, rounding your answer to three decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Compute the momentum of one $500 \mathrm{~nm}$ photon using $p_{\text {photon }}=E_{\text {photon }} / c$ where $c$ is the speed of light, $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$, and $\nu=c / \lambda$.  Express your answer in kilogram meters per second, rounding your answer to three decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Compute the momentum of one $500 \mathrm{~nm}$ photon using $p_{\text {photon }}=E_{\text {photon }} / c$ where $c$ is the speed of light, $c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$, and $\nu=c / \lambda$.  Express your answer in kilogram meters per second, rounding your answer to three decimal places.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem deals with the H\"uckel MO theory of $\pi$-conjugated systems.
To answer each question, you will need to construct the Hückel MOs for each of the molecules pictured, divide them into sets of occupied and unoccupied orbitals, and determine the relevant properties, such as ground state energy, bond order, etc.
NOTE: For all parts we take $\alpha=\alpha_{\mathrm{C}}=-11.2 \mathrm{eV}$ and $\beta=\beta_{\mathrm{CC}}=-0.7 \mathrm{eV}$.

Determine the ionization potential of benzene (remember, ionization potential $\left[\mathrm{IP}=\mathrm{E}\left(\mathrm{B}^{+}\right)-\mathrm{E}(\mathrm{B})\right]$), in $\mathrm{eV}$, rounded to one decimal place.  The benzene molecule is shown below:
\chemfig{C*6((-H)-C(-H)=C(-H)-C(-H)=C(-H)-C(-H)=)}
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem deals with the H\"uckel MO theory of $\pi$-conjugated systems.
To answer each question, you will need to construct the Hückel MOs for each of the molecules pictured, divide them into sets of occupied and unoccupied orbitals, and determine the relevant properties, such as ground state energy, bond order, etc.
NOTE: For all parts we take $\alpha=\alpha_{\mathrm{C}}=-11.2 \mathrm{eV}$ and $\beta=\beta_{\mathrm{CC}}=-0.7 \mathrm{eV}$.

Determine the ionization potential of benzene (remember, ionization potential $\left[\mathrm{IP}=\mathrm{E}\left(\mathrm{B}^{+}\right)-\mathrm{E}(\mathrm{B})\right]$), in $\mathrm{eV}$, rounded to one decimal place.  The benzene molecule is shown below:
\chemfig{C*6((-H)-C(-H)=C(-H)-C(-H)=C(-H)-C(-H)=)}
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem deals with the H\"uckel MO theory of $\pi$-conjugated systems.
To answer each question, you will need to construct the Hückel MOs for each of the molecules pictured, divide them into sets of occupied and unoccupied orbitals, and determine the relevant properties, such as ground state energy, bond order, etc.
NOTE: For all parts we take $\alpha=\alpha_{\mathrm{C}}=-11.2 \mathrm{eV}$ and $\beta=\beta_{\mathrm{CC}}=-0.7 \mathrm{eV}$.

Determine the ionization potential of benzene (remember, ionization potential $\left[\mathrm{IP}=\mathrm{E}\left(\mathrm{B}^{+}\right)-\mathrm{E}(\mathrm{B})\right]$), in $\mathrm{eV}$, rounded to one decimal place.  The benzene molecule is shown below:
\chemfig{C*6((-H)-C(-H)=C(-H)-C(-H)=C(-H)-C(-H)=)}
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: This problem deals with the H\"uckel MO theory of $\pi$-conjugated systems.
To answer each question, you will need to construct the Hückel MOs for each of the molecules pictured, divide them into sets of occupied and unoccupied orbitals, and determine the relevant properties, such as ground state energy, bond order, etc.
NOTE: For all parts we take $\alpha=\alpha_{\mathrm{C}}=-11.2 \mathrm{eV}$ and $\beta=\beta_{\mathrm{CC}}=-0.7 \mathrm{eV}$.

Determine the ionization potential of benzene (remember, ionization potential $\left[\mathrm{IP}=\mathrm{E}\left(\mathrm{B}^{+}\right)-\mathrm{E}(\mathrm{B})\right]$), in $\mathrm{eV}$, rounded to one decimal place.  The benzene molecule is shown below:
\chemfig{C*6((-H)-C(-H)=C(-H)-C(-H)=C(-H)-C(-H)=)}
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A baseball has diameter $=7.4 \mathrm{~cm}$. and a mass of $145 \mathrm{~g}$. Suppose the baseball is moving at $v=1 \mathrm{~nm} /$ second. What is its de Broglie wavelength
\[
\lambda=\frac{h}{p}=\frac{h}{m \nu}
\]
?  Give answer in meters.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A baseball has diameter $=7.4 \mathrm{~cm}$. and a mass of $145 \mathrm{~g}$. Suppose the baseball is moving at $v=1 \mathrm{~nm} /$ second. What is its de Broglie wavelength
\[
\lambda=\frac{h}{p}=\frac{h}{m \nu}
\]
?  Give answer in meters.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A baseball has diameter $=7.4 \mathrm{~cm}$. and a mass of $145 \mathrm{~g}$. Suppose the baseball is moving at $v=1 \mathrm{~nm} /$ second. What is its de Broglie wavelength
\[
\lambda=\frac{h}{p}=\frac{h}{m \nu}
\]
?  Give answer in meters.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
A baseball has diameter $=7.4 \mathrm{~cm}$. and a mass of $145 \mathrm{~g}$. Suppose the baseball is moving at $v=1 \mathrm{~nm} /$ second. What is its de Broglie wavelength
\[
\lambda=\frac{h}{p}=\frac{h}{m \nu}
\]
?  Give answer in meters.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Subproblem 0: Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.

Solution: Since the only eigenergies are $E_{1}$ and $E_{2}$, the possible outcomes of the measurement are $\boxed{E_{1},E_{2}}$.

Final answer: The final answer is E_{1},E_{2}. I hope it is correct.

Subproblem 1: Suppose you do many identical measurements to measure the energies of identical systems in state $\psi_{1,2}$. What average energy will you observe?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Subproblem 0: Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.

Solution: Since the only eigenergies are $E_{1}$ and $E_{2}$, the possible outcomes of the measurement are $\boxed{E_{1},E_{2}}$.

Final answer: The final answer is E_{1},E_{2}. I hope it is correct.

Subproblem 1: Suppose you do many identical measurements to measure the energies of identical systems in state $\psi_{1,2}$. What average energy will you observe?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Subproblem 0: Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.

Solution: Since the only eigenergies are $E_{1}$ and $E_{2}$, the possible outcomes of the measurement are $\boxed{E_{1},E_{2}}$.

Final answer: The final answer is E_{1},E_{2}. I hope it is correct.

Subproblem 1: Suppose you do many identical measurements to measure the energies of identical systems in state $\psi_{1,2}$. What average energy will you observe?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Subproblem 0: Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.

Solution: Since the only eigenergies are $E_{1}$ and $E_{2}$, the possible outcomes of the measurement are $\boxed{E_{1},E_{2}}$.

Final answer: The final answer is E_{1},E_{2}. I hope it is correct.

Subproblem 1: Suppose you do many identical measurements to measure the energies of identical systems in state $\psi_{1,2}$. What average energy will you observe?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the Particle in an Infinite Box ``superposition state'' wavefunction,
\[
\psi_{1,2}=(1 / 3)^{1 / 2} \psi_{1}+(2 / 3)^{1 / 2} \psi_{2}
\]
where $E_{1}$ is the eigen-energy of $\psi_{1}$ and $E_{2}$ is the eigen-energy of $\psi_{2}$.

Suppose you do one experiment to measure the energy of $\psi_{1,2}$.  List the possible result(s) of your measurement.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Evaluate the following integrals for $\psi_{J M}$ eigenfunctions of $\mathbf{J}^{2}$ and $\mathbf{J}_{z}$.

$\int \psi_{22}^{*}\left(\widehat{\mathbf{J}}^{+}\right)^{4} \psi_{2,-2} d \tau$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Evaluate the following integrals for $\psi_{J M}$ eigenfunctions of $\mathbf{J}^{2}$ and $\mathbf{J}_{z}$.

$\int \psi_{22}^{*}\left(\widehat{\mathbf{J}}^{+}\right)^{4} \psi_{2,-2} d \tau$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Evaluate the following integrals for $\psi_{J M}$ eigenfunctions of $\mathbf{J}^{2}$ and $\mathbf{J}_{z}$.

$\int \psi_{22}^{*}\left(\widehat{\mathbf{J}}^{+}\right)^{4} \psi_{2,-2} d \tau$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Evaluate the following integrals for $\psi_{J M}$ eigenfunctions of $\mathbf{J}^{2}$ and $\mathbf{J}_{z}$.

$\int \psi_{22}^{*}\left(\widehat{\mathbf{J}}^{+}\right)^{4} \psi_{2,-2} d \tau$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the 3-level $\mathbf{H}$ matrix
\[
\mathbf{H}=\hbar \omega\left(\begin{array}{ccc}
10 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -10
\end{array}\right)
\]
Label the eigen-energies and eigen-functions according to the dominant basis state character. The $\widetilde{10}$ state is the one dominated by the zero-order state with $E^{(0)}=10, \tilde{0}$ by $E^{(0)}=0$, and $-\widetilde{10}$ by $E^{(0)}=-10$ (we will work in units where $\hbar \omega = 1$, and can be safely ignored).

Use non-degenerate perturbation theory to derive the energy $E_{\widetilde{10}}$.  Carry out your calculations to second order in the perturbing Hamiltonian, and round to one decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the 3-level $\mathbf{H}$ matrix
\[
\mathbf{H}=\hbar \omega\left(\begin{array}{ccc}
10 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -10
\end{array}\right)
\]
Label the eigen-energies and eigen-functions according to the dominant basis state character. The $\widetilde{10}$ state is the one dominated by the zero-order state with $E^{(0)}=10, \tilde{0}$ by $E^{(0)}=0$, and $-\widetilde{10}$ by $E^{(0)}=-10$ (we will work in units where $\hbar \omega = 1$, and can be safely ignored).

Use non-degenerate perturbation theory to derive the energy $E_{\widetilde{10}}$.  Carry out your calculations to second order in the perturbing Hamiltonian, and round to one decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the 3-level $\mathbf{H}$ matrix
\[
\mathbf{H}=\hbar \omega\left(\begin{array}{ccc}
10 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -10
\end{array}\right)
\]
Label the eigen-energies and eigen-functions according to the dominant basis state character. The $\widetilde{10}$ state is the one dominated by the zero-order state with $E^{(0)}=10, \tilde{0}$ by $E^{(0)}=0$, and $-\widetilde{10}$ by $E^{(0)}=-10$ (we will work in units where $\hbar \omega = 1$, and can be safely ignored).

Use non-degenerate perturbation theory to derive the energy $E_{\widetilde{10}}$.  Carry out your calculations to second order in the perturbing Hamiltonian, and round to one decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant


Prompt:
<|im_start|>user
Preamble: Consider the 3-level $\mathbf{H}$ matrix
\[
\mathbf{H}=\hbar \omega\left(\begin{array}{ccc}
10 & 1 & 0 \\
1 & 0 & 2 \\
0 & 2 & -10
\end{array}\right)
\]
Label the eigen-energies and eigen-functions according to the dominant basis state character. The $\widetilde{10}$ state is the one dominated by the zero-order state with $E^{(0)}=10, \tilde{0}$ by $E^{(0)}=0$, and $-\widetilde{10}$ by $E^{(0)}=-10$ (we will work in units where $\hbar \omega = 1$, and can be safely ignored).

Use non-degenerate perturbation theory to derive the energy $E_{\widetilde{10}}$.  Carry out your calculations to second order in the perturbing Hamiltonian, and round to one decimal place.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant