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Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Calculus HW
![### Calculating Revenue and Profit
**Price Function:**
\[ p(x) = 400 - 5x \]
**Objective:**
Find revenue when \( x = 12 \).
**Revenue Calculation:**
Revenue is given by the function:
\[ \text{Revenue} = P(x) \cdot x \]
Substituting for \( P(x) \):
\[ (400 - 5x) \cdot x \]
This simplifies to:
\[ R(x) = 400x - 5x^2 \]
**Product Calculation:**
Define Product as the difference between Revenue and Cost:
\[ \text{Product} = R(x) - C(x) \]
Given:
\[ P(x) = 400x - 5x^2 - 12 \]
**Finding the Derivative of Product:**
\[ P'(x) = 400 - 10x \]
**Finding Critical Points:**
Set the derivative to zero to find critical points:
\[ P'(x) = 0 \]
\[ 400 - 10x = 0 \]
Solving for \( x \):
\[ 400 = 10x \]
\[ \frac{400}{10} = x \]
\[ 40 = x \]
Thus, the critical point occurs at \( x = 40 \).](https://content.bartleby.com/qna-images/question/eb62ec4b-b625-4332-acd2-c86081f32c3c/e794e1ab-2473-4031-98a4-68cda14a2d0e/8hh75v_thumbnail.jpeg)
Transcribed Image Text:### Calculating Revenue and Profit
**Price Function:**
\[ p(x) = 400 - 5x \]
**Objective:**
Find revenue when \( x = 12 \).
**Revenue Calculation:**
Revenue is given by the function:
\[ \text{Revenue} = P(x) \cdot x \]
Substituting for \( P(x) \):
\[ (400 - 5x) \cdot x \]
This simplifies to:
\[ R(x) = 400x - 5x^2 \]
**Product Calculation:**
Define Product as the difference between Revenue and Cost:
\[ \text{Product} = R(x) - C(x) \]
Given:
\[ P(x) = 400x - 5x^2 - 12 \]
**Finding the Derivative of Product:**
\[ P'(x) = 400 - 10x \]
**Finding Critical Points:**
Set the derivative to zero to find critical points:
\[ P'(x) = 0 \]
\[ 400 - 10x = 0 \]
Solving for \( x \):
\[ 400 = 10x \]
\[ \frac{400}{10} = x \]
\[ 40 = x \]
Thus, the critical point occurs at \( x = 40 \).
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