xミ12、 %3D Renenue = P(x).X (400-ち×)× Rx)= 4wx - 5x2 find Revenue when x=12

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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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 \( x = 40 \).
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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