of copies Foothill Bookstore can sell each week if it sets the price a when the price is set to $40 per copy. SH- dp q A) Find the price elasticity of demand, E = -Ca) = C400-8)" 100 9² = Too — (400-p3² 160 =1 (00 .2 (400-р.). ep 800-29 (0-12 = 100 1-800-20 100 800-100 (400-5) q= (400-402² 36 C 0.0100 (0
of copies Foothill Bookstore can sell each week if it sets the price a when the price is set to $40 per copy. SH- dp q A) Find the price elasticity of demand, E = -Ca) = C400-8)" 100 9² = Too — (400-p3² 160 =1 (00 .2 (400-р.). ep 800-29 (0-12 = 100 1-800-20 100 800-100 (400-5) q= (400-402² 36 C 0.0100 (0
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Answer A, B, and C, with the information provided. Show all work.
![### Part IV—Elasticity
#### 1. The demand curve for the graphic novel *Night of the Owl* is given by:
\[ q = \frac{(400 - p^2)}{100} \; (0 \leq p \leq 400) \]
where \( q \) is the number of copies Foothill Bookstore can sell each week if it sets the price at \( p \) dollars.
#### A) Find the price elasticity of demand, \( E = -\frac{dq}{dp} \cdot \frac{p}{q} \) when the price is set to $40 per copy. SHOW WORK.
---
#### Solution:
1. **Differentiate the demand function with respect to price \( p \)**:
\[ q = \frac{400 - p^2}{100} \]
\[ \frac{dq}{dp} = \frac{d}{dp} \left( \frac{400 - p^2}{100} \right) = \frac{d}{dp} \left( \frac{C(400 - p^2)}{100} \right) \]
\[ = \frac{1}{100} \cdot \left( 0 - 2p \right) \]
\[ = \frac{-2p}{100} \]
\[ = -\frac{2p}{100} \]
2. **Calculate \( q \) when \( p = 40 \)**:
\[ q = \frac{(400 - 40^2)}{100} \]
\[ = \frac{(400 - 1600)}{100} \]
\[ = \frac{(3600)}{100} \]
\[ = \frac{129600}{100} = 1296 \]
3. **Compute elasticity \( E \)**:
\[ E = -\frac{dq}{dp} \cdot \frac{p}{q} \]
\[ E = -\left( \frac{-2p}{100} \right) \cdot \frac{p}{q} \]
\[ = \left( \frac{2 \cdot 40}{100} \right) \cdot \frac{40}{1296} \]
\[ = \left( \frac{80}{100} \cdot \frac{40}{1296} \right) \]
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2e37893-b1d8-4cac-bb15-a057cf9332c6%2F7a9084d6-dc7e-41f3-b10b-3a224063f0fd%2Fxgpgee4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Part IV—Elasticity
#### 1. The demand curve for the graphic novel *Night of the Owl* is given by:
\[ q = \frac{(400 - p^2)}{100} \; (0 \leq p \leq 400) \]
where \( q \) is the number of copies Foothill Bookstore can sell each week if it sets the price at \( p \) dollars.
#### A) Find the price elasticity of demand, \( E = -\frac{dq}{dp} \cdot \frac{p}{q} \) when the price is set to $40 per copy. SHOW WORK.
---
#### Solution:
1. **Differentiate the demand function with respect to price \( p \)**:
\[ q = \frac{400 - p^2}{100} \]
\[ \frac{dq}{dp} = \frac{d}{dp} \left( \frac{400 - p^2}{100} \right) = \frac{d}{dp} \left( \frac{C(400 - p^2)}{100} \right) \]
\[ = \frac{1}{100} \cdot \left( 0 - 2p \right) \]
\[ = \frac{-2p}{100} \]
\[ = -\frac{2p}{100} \]
2. **Calculate \( q \) when \( p = 40 \)**:
\[ q = \frac{(400 - 40^2)}{100} \]
\[ = \frac{(400 - 1600)}{100} \]
\[ = \frac{(3600)}{100} \]
\[ = \frac{129600}{100} = 1296 \]
3. **Compute elasticity \( E \)**:
\[ E = -\frac{dq}{dp} \cdot \frac{p}{q} \]
\[ E = -\left( \frac{-2p}{100} \right) \cdot \frac{p}{q} \]
\[ = \left( \frac{2 \cdot 40}{100} \right) \cdot \frac{40}{1296} \]
\[ = \left( \frac{80}{100} \cdot \frac{40}{1296} \right) \]
\[
![### Educational Exercise on Price Elasticity and Revenue Optimization
**Instructions:**
This section includes a series of questions designed to test your understanding of price elasticity and its impact on revenue optimization. Read each question carefully and provide comprehensive answers.
#### Question B:
**B) Is this price elastic or inelastic? Should the bookstore charge more or less to maximize revenue? Explain.**
**Answer:**
- *This price is inelastic.*
#### Question C:
**C) What is the elasticity of the price that maximizes revenue? Write an equation using this value whose solution gives the price maximizing weekly revenue for the bookstore. SHOW WORK.**
**Answer:**
- To determine the elasticity of the price that maximizes revenue, we need to use the concept of elasticity which is generally defined as:
\[
E = \frac{\partial Q / Q}{\partial P / P}
\]
where \(E\) is the elasticity.
Let's assume there is a linear demand curve given by the equation:
\[
Q = a - bP
\]
Therefore, Total Revenue \(TR\) is:
\[
TR = P \times Q = P (a - bP)
\]
To find the price that maximizes revenue, differentiate \(TR\) with respect to \(P\) and set the derivative to zero:
\[
\frac{d(TR)}{dP} = a - 2bP = 0
\]
Solving for \(P\), we get:
\[
P = \frac{a}{2b}
\]
#### Question D:
**D) What is the maximum weekly revenue? SHOW WORK and give the final answer in sentence form.**
**Answer:**
- Substitute \(P = \frac{a}{2b}\) back into the total revenue equation to find the maximum revenue:
\[
TR = P \times Q = \left(\frac{a}{2b}\right) \times \left(a - b \left(\frac{a}{2b}\right) \right)
\]
Simplify inside the parentheses first:
\[
Q = a - b \left(\frac{a}{2b}\right) = a - \frac{a}{2} = \frac{a}{2}
\]
Thus,
\[
TR](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2e37893-b1d8-4cac-bb15-a057cf9332c6%2F7a9084d6-dc7e-41f3-b10b-3a224063f0fd%2Fa83cs8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Exercise on Price Elasticity and Revenue Optimization
**Instructions:**
This section includes a series of questions designed to test your understanding of price elasticity and its impact on revenue optimization. Read each question carefully and provide comprehensive answers.
#### Question B:
**B) Is this price elastic or inelastic? Should the bookstore charge more or less to maximize revenue? Explain.**
**Answer:**
- *This price is inelastic.*
#### Question C:
**C) What is the elasticity of the price that maximizes revenue? Write an equation using this value whose solution gives the price maximizing weekly revenue for the bookstore. SHOW WORK.**
**Answer:**
- To determine the elasticity of the price that maximizes revenue, we need to use the concept of elasticity which is generally defined as:
\[
E = \frac{\partial Q / Q}{\partial P / P}
\]
where \(E\) is the elasticity.
Let's assume there is a linear demand curve given by the equation:
\[
Q = a - bP
\]
Therefore, Total Revenue \(TR\) is:
\[
TR = P \times Q = P (a - bP)
\]
To find the price that maximizes revenue, differentiate \(TR\) with respect to \(P\) and set the derivative to zero:
\[
\frac{d(TR)}{dP} = a - 2bP = 0
\]
Solving for \(P\), we get:
\[
P = \frac{a}{2b}
\]
#### Question D:
**D) What is the maximum weekly revenue? SHOW WORK and give the final answer in sentence form.**
**Answer:**
- Substitute \(P = \frac{a}{2b}\) back into the total revenue equation to find the maximum revenue:
\[
TR = P \times Q = \left(\frac{a}{2b}\right) \times \left(a - b \left(\frac{a}{2b}\right) \right)
\]
Simplify inside the parentheses first:
\[
Q = a - b \left(\frac{a}{2b}\right) = a - \frac{a}{2} = \frac{a}{2}
\]
Thus,
\[
TR
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