The total cost (in dollars) of producing x food processors is C(x) = 1600+50x-0.4x². Find the marginal cost funtion and use it to approximate the cost of producing the 31st food processor. .... The approximate cost of producing the 31st food processor is $.
The total cost (in dollars) of producing x food processors is C(x) = 1600+50x-0.4x². Find the marginal cost funtion and use it to approximate the cost of producing the 31st food processor. .... The approximate cost of producing the 31st food processor is $.
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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![### Calculating Marginal Cost in Production
The total cost \( C(x) \) (in dollars) of producing \( x \) food processors is given by the function:
\[ C(x) = 1600 + 50x - 0.4x^2 \]
To find the marginal cost function, which represents the cost of producing one additional unit, we need to determine the derivative of the total cost function \( C(x) \).
**Marginal Cost Function:**
1. Given the cost function:
\[ C(x) = 1600 + 50x - 0.4x^2 \]
2. Differentiate \( C(x) \) with respect to \( x \):
\[ C'(x) = \frac{d}{dx} (1600 + 50x - 0.4x^2) \]
\[ C'(x) = 0 + 50 - 0.8x \]
\[ C'(x) = 50 - 0.8x \]
The marginal cost function is:
\[ C'(x) = 50 - 0.8x \]
**Approximation of the Cost for the 31st Food Processor:**
To approximate the cost of producing the 31st food processor, substitute \( x = 31 \) into the marginal cost function:
\[ C'(31) = 50 - 0.8(31) \]
\[ C'(31) = 50 - 24.8 \]
\[ C'(31) = 25.2 \]
Therefore, the approximate cost of producing the 31st food processor is:
\[ \$25.20 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11a64bba-74c0-4fd7-b16b-3d8fff0a6ac2%2F9fbc109d-88bd-467e-be93-dced62be98aa%2Fj16obt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating Marginal Cost in Production
The total cost \( C(x) \) (in dollars) of producing \( x \) food processors is given by the function:
\[ C(x) = 1600 + 50x - 0.4x^2 \]
To find the marginal cost function, which represents the cost of producing one additional unit, we need to determine the derivative of the total cost function \( C(x) \).
**Marginal Cost Function:**
1. Given the cost function:
\[ C(x) = 1600 + 50x - 0.4x^2 \]
2. Differentiate \( C(x) \) with respect to \( x \):
\[ C'(x) = \frac{d}{dx} (1600 + 50x - 0.4x^2) \]
\[ C'(x) = 0 + 50 - 0.8x \]
\[ C'(x) = 50 - 0.8x \]
The marginal cost function is:
\[ C'(x) = 50 - 0.8x \]
**Approximation of the Cost for the 31st Food Processor:**
To approximate the cost of producing the 31st food processor, substitute \( x = 31 \) into the marginal cost function:
\[ C'(31) = 50 - 0.8(31) \]
\[ C'(31) = 50 - 24.8 \]
\[ C'(31) = 25.2 \]
Therefore, the approximate cost of producing the 31st food processor is:
\[ \$25.20 \]
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