A sales representative for a company that produces blenders can sell x units of their deluxe model if the price is = 0.08x2 +6.9x + 5700 dollars. p = D(x) = 83.4 -0.02x dollars. The total cost for these blenders is given by C(x) Determine the marginal profit for 134 blenders.
A sales representative for a company that produces blenders can sell x units of their deluxe model if the price is = 0.08x2 +6.9x + 5700 dollars. p = D(x) = 83.4 -0.02x dollars. The total cost for these blenders is given by C(x) Determine the marginal profit for 134 blenders.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter3: Linear And Nonlinear Functions
Section: Chapter Questions
Problem 26MCQ
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![### Marginal Profit Calculation
A sales representative for a company that produces blenders can sell \( x \) units of their deluxe model if the price is
\[ p = D(x) = 83.4 - 0.02x \text{ dollars.} \]
The total cost for these blenders is given by
\[ C(x) = 0.08x^2 + 6.9x + 5700 \text{ dollars.} \]
#### Problem Statement:
Determine the marginal profit for 134 blenders.
#### Explanation:
To find the marginal profit, we need to calculate the derivative of the profit function \( P(x) \) with respect to \( x \) evaluated at \( x = 134 \).
1. **Revenue Function \( R(x) \):**
The revenue function \( R(x) \) is price \( p \) times the number of units sold \( x \):
\[ R(x) = x \cdot p = x \cdot (83.4 - 0.02x) = 83.4x - 0.02x^2 \]
2. **Cost Function \( C(x) \) (given):**
\[ C(x) = 0.08x^2 + 6.9x + 5700 \]
3. **Profit Function \( P(x) \):**
The profit function is the revenue function minus the cost function:
\[ P(x) = R(x) - C(x) = (83.4x - 0.02x^2) - (0.08x^2 + 6.9x + 5700) \]
Simplifying, we get:
\[ P(x) = 83.4x - 0.02x^2 - 0.08x^2 - 6.9x - 5700 \]
\[ P(x) = 83.4x - 6.9x - 0.1x^2 - 5700 \]
\[ P(x) = 76.5x - 0.1x^2 - 5700 \]
4. **Marginal Profit \( P'(x) \):**
To find the rate of change of profit with respect to the number of units, take the derivative:
\[ P'(x) = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F323c4818-e3ec-479b-a7ba-1a514193e117%2Fd7524145-c978-43a3-93e4-e299a75dc2a0%2F2j8sgnw_processed.png&w=3840&q=75)
Transcribed Image Text:### Marginal Profit Calculation
A sales representative for a company that produces blenders can sell \( x \) units of their deluxe model if the price is
\[ p = D(x) = 83.4 - 0.02x \text{ dollars.} \]
The total cost for these blenders is given by
\[ C(x) = 0.08x^2 + 6.9x + 5700 \text{ dollars.} \]
#### Problem Statement:
Determine the marginal profit for 134 blenders.
#### Explanation:
To find the marginal profit, we need to calculate the derivative of the profit function \( P(x) \) with respect to \( x \) evaluated at \( x = 134 \).
1. **Revenue Function \( R(x) \):**
The revenue function \( R(x) \) is price \( p \) times the number of units sold \( x \):
\[ R(x) = x \cdot p = x \cdot (83.4 - 0.02x) = 83.4x - 0.02x^2 \]
2. **Cost Function \( C(x) \) (given):**
\[ C(x) = 0.08x^2 + 6.9x + 5700 \]
3. **Profit Function \( P(x) \):**
The profit function is the revenue function minus the cost function:
\[ P(x) = R(x) - C(x) = (83.4x - 0.02x^2) - (0.08x^2 + 6.9x + 5700) \]
Simplifying, we get:
\[ P(x) = 83.4x - 0.02x^2 - 0.08x^2 - 6.9x - 5700 \]
\[ P(x) = 83.4x - 6.9x - 0.1x^2 - 5700 \]
\[ P(x) = 76.5x - 0.1x^2 - 5700 \]
4. **Marginal Profit \( P'(x) \):**
To find the rate of change of profit with respect to the number of units, take the derivative:
\[ P'(x) = \
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