An accessories company finds that the​ cost, in​ dollars, of producing x belts is given by

Upper C left parenthesis x right parenthesis equals 720 plus 31 x minus 0.063 x squaredC(x)=720+31x−0.063x2.

Find the rate at which average cost is changing when

174174

belts have been produced.

Question content area bottom

Part 1

​First, find the rate at which the average cost is changing when x belts have been produced.

To find the rate at which the average cost is changing, we first need to calculate the **average cost function** \( A(x) \), which is given by dividing the total cost function \( C(x) \) by \( x \).

Given:
\[
C(x) = 720 + 31x - 0.063x^2
\]
The average cost function \( A(x) \) is:
\[
A(x) = \frac{C(x)}{x} = \frac{720 + 31x - 0.063x^2}{x}
\]
Simplifying the right-hand side:
\[
A(x) = \frac{720}{x} + 31 - 0.063x
\]

Now, we want to find the **rate of change** of the average cost, which is the derivative of \( A(x) \) with respect to \( x \). So, we differentiate \( A(x) \):

\[
A'(x) = \frac{d}{dx} \left( \frac{720}{x} + 31 - 0.063x \right)
\]

Differentiating each term:
\[
A'(x) = -\frac{720}{x^2} + 0 - 0.063 = -\frac{720}{x^2} - 0.063
\]

This expression represents the rate of change of the average cost when \( x \) belts have been produced.

### Part 2: Calculate for 174 belts

To find the rate of change of the average cost when 174 belts have been produced, we substitute \( x = 174 \) into \( A'(x) \):

\[
A'(174) = -\frac{720}{174^2} - 0.063
\]

Now, let's calculate that value.The rate at which the average cost is changing when 174 belts have been produced is approximately \(-0.087\) dollars per belt. This means the average cost is decreasing at a rate of about 8.7 cents per belt when 174 belts are produced.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How does the total cost change when more belts are produced?
2. What is the average cost per belt when producing 174 belts?
3. How does the average cost function behave as the number of belts increases?
4. Can the rate of change in average cost be zero, and if so, at what number of belts?
5. What is the minimum possible average cost, if it exists?

**Tip:** The average cost function helps in determining the optimal production quantity to minimize the cost per unit.

An accessories company finds that the cost, in dollars, of producing x belts is given by C(x) = 720 + 31x - 0.063x^2. Find the rate at which average cost is changing when 174 belts have been produced.

Discover how to calculate the rate at which the average cost is changing for producing belts using the cost function C(x) = 720 + 31x - 0.063x^2. This problem explores calculus concepts like average cost and derivatives, and is ideal for Grades 11-12 or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation