Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 6DE
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![**Consumer Surplus Calculation Example**
This example demonstrates how to find the consumer surplus at a specific price level, utilizing a given price-demand equation. Consider the following problem:
---
**Problem Statement:**
Find the consumers' surplus at a price level of \( p = $150 \) for the price-demand equation below:
\[ p = D(x) = 400 - 0.02x \]
**Question:**
What is the consumer surplus?
\[ \$ \_\_\_\_\_\_ \]
---
**Solution:**
To calculate the consumer surplus, follow these steps:
1. **Set up the equation** to find the quantity demanded (\( x \)) at the price level \( p = $150 \).
\[ 150 = 400 - 0.02x \]
2. **Solve for \( x \):**
\[
150 = 400 - 0.02x
\]
\[
0.02x = 400 - 150
\]
\[
0.02x = 250
\]
\[
x = \frac{250}{0.02}
\]
\[
x = 12500
\]
3. **Find the maximum price consumers are willing to pay:** This is the price at \( x = 0 \):
\[
p = 400 - 0.02(0) = 400
\]
4. **Integrate the demand function** from 0 to 12500 to find the total willingness to pay:
\[
\int_{0}^{12500} (400 - 0.02x) \, dx
\]
5. **Calculate the integral** to determine the area under the demand curve:
\[
\int_{0}^{12500} 400 \, dx - \int_{0}^{12500} 0.02x \, dx
\]
\[
= 400x \bigg|_0^{12500} - 0.02 \cdot \frac{x^2}{2} \bigg|_0^{12500}
\]
\[
= 400(12500) - 0.02 \cdot \frac{(12500)^2}{2}
\]
\[
= 5000000 -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e7a21da-f67b-40d7-b2d8-1c29d4472d02%2F91536a7b-1f9b-487f-a6e6-b6c30042293b%2Fd84e00l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consumer Surplus Calculation Example**
This example demonstrates how to find the consumer surplus at a specific price level, utilizing a given price-demand equation. Consider the following problem:
---
**Problem Statement:**
Find the consumers' surplus at a price level of \( p = $150 \) for the price-demand equation below:
\[ p = D(x) = 400 - 0.02x \]
**Question:**
What is the consumer surplus?
\[ \$ \_\_\_\_\_\_ \]
---
**Solution:**
To calculate the consumer surplus, follow these steps:
1. **Set up the equation** to find the quantity demanded (\( x \)) at the price level \( p = $150 \).
\[ 150 = 400 - 0.02x \]
2. **Solve for \( x \):**
\[
150 = 400 - 0.02x
\]
\[
0.02x = 400 - 150
\]
\[
0.02x = 250
\]
\[
x = \frac{250}{0.02}
\]
\[
x = 12500
\]
3. **Find the maximum price consumers are willing to pay:** This is the price at \( x = 0 \):
\[
p = 400 - 0.02(0) = 400
\]
4. **Integrate the demand function** from 0 to 12500 to find the total willingness to pay:
\[
\int_{0}^{12500} (400 - 0.02x) \, dx
\]
5. **Calculate the integral** to determine the area under the demand curve:
\[
\int_{0}^{12500} 400 \, dx - \int_{0}^{12500} 0.02x \, dx
\]
\[
= 400x \bigg|_0^{12500} - 0.02 \cdot \frac{x^2}{2} \bigg|_0^{12500}
\]
\[
= 400(12500) - 0.02 \cdot \frac{(12500)^2}{2}
\]
\[
= 5000000 -
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