Find the producers' surplus at a price level of p = $61 for the price-supply equation below. p= S(x)=15+0.1x+0.0003x² The producers' surplus is $. (Round to the nearest integer as needed.)
Find the producers' surplus at a price level of p = $61 for the price-supply equation below. p= S(x)=15+0.1x+0.0003x² The producers' surplus is $. (Round to the nearest integer as needed.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Finding the Producer's Surplus**
**Objective:** Determine the producers' surplus at a price level of \( \bar{p} = \$61 \) for the given price-supply equation.
Given price-supply equation:
\[ p = S(x) = 15 + 0.1x + 0.0003x^2 \]
1. **Understanding the problem:**
- We need to find the producer's surplus, which is the difference between what producers are willing to accept for selling their product versus what they actually receive.
- **Producer's Surplus Formula:**
\[ \text{Producer Surplus (PS)} = \int_0^{x*} S(x) \, dx - x* \cdot \bar{p} \]
Where \( x* \) is the quantity supplied when the price \( \bar{p} = \$61 \).
2. **Solve for \( x* \) when \( \bar{p} = \$61 \):**
- Substitute \(\bar{p} = 61\) into the price-supply equation:
\[ 61 = 15 + 0.1x + 0.0003x^2 \]
- Solve for \( x \):
\[ 46 = 0.1x + 0.0003x^2 \]
\[ 0.0003x^2 + 0.1x - 46 = 0 \]
3. **Calculate the definite integral of the supply function \( S(x) \) from 0 to \( x* \)**:
- Find the antiderivative \( \int S(x) \, dx \):
\[ \int (15 + 0.1x + 0.0003x^2) \, dx = 15x + 0.05x^2 + 0.0001x^3 \]
- Evaluate this antiderivative from 0 to \( x* \).
4. **Compute the producer's surplus**:
- Substitute the values obtained from the definite integral and product of \( x* \) and \( \bar{p} \) into the producer surplus formula to find the final value, and round to the nearest integer as needed.
**Please Note:** The detailed calculations for solving the quadratic equation and evaluating the integral are essential steps but are omitted](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e7a21da-f67b-40d7-b2d8-1c29d4472d02%2Fd7d9eea5-e042-4449-95ed-5b68d26793c7%2F82rm283_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Producer's Surplus**
**Objective:** Determine the producers' surplus at a price level of \( \bar{p} = \$61 \) for the given price-supply equation.
Given price-supply equation:
\[ p = S(x) = 15 + 0.1x + 0.0003x^2 \]
1. **Understanding the problem:**
- We need to find the producer's surplus, which is the difference between what producers are willing to accept for selling their product versus what they actually receive.
- **Producer's Surplus Formula:**
\[ \text{Producer Surplus (PS)} = \int_0^{x*} S(x) \, dx - x* \cdot \bar{p} \]
Where \( x* \) is the quantity supplied when the price \( \bar{p} = \$61 \).
2. **Solve for \( x* \) when \( \bar{p} = \$61 \):**
- Substitute \(\bar{p} = 61\) into the price-supply equation:
\[ 61 = 15 + 0.1x + 0.0003x^2 \]
- Solve for \( x \):
\[ 46 = 0.1x + 0.0003x^2 \]
\[ 0.0003x^2 + 0.1x - 46 = 0 \]
3. **Calculate the definite integral of the supply function \( S(x) \) from 0 to \( x* \)**:
- Find the antiderivative \( \int S(x) \, dx \):
\[ \int (15 + 0.1x + 0.0003x^2) \, dx = 15x + 0.05x^2 + 0.0001x^3 \]
- Evaluate this antiderivative from 0 to \( x* \).
4. **Compute the producer's surplus**:
- Substitute the values obtained from the definite integral and product of \( x* \) and \( \bar{p} \) into the producer surplus formula to find the final value, and round to the nearest integer as needed.
**Please Note:** The detailed calculations for solving the quadratic equation and evaluating the integral are essential steps but are omitted
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