Assume that it costs **Apple** approximately *C*(*x*) =32,400+ 100*x* + 0.01*x*2 dollars to manufacture *x* 30-gigabyte video iPods in a day.† How many iPods should be manufactured in order to minimize average cost? (Round to the nearest hundred.)

What is the resulting average cost of an iPod? (Give your answer to the nearest dollar.) HINT [See Example 1.]

We are given the cost function $ C(x) = 32,400 + 100x + 0.01x^2 $, which represents the total cost to produce $ x $ iPods in a day. To minimize the average cost, we need to find the number of iPods that minimizes the average cost per iPod.

### Steps to solve:

1. **Average cost function:**
   The average cost function $ AC(x) $ is given by:
   \[
   AC(x) = \frac{C(x)}{x}
   \]
   Substituting $ C(x) = 32,400 + 100x + 0.01x^2 $ into this equation:
   \[
   AC(x) = \frac{32,400 + 100x + 0.01x^2}{x} = \frac{32,400}{x} + 100 + 0.01x
   \]

2. **Minimizing the average cost:**
   To find the number of iPods $ x $ that minimizes the average cost, we need to take the derivative of $ AC(x) $ and set it equal to zero.

   First, differentiate $ AC(x) $ with respect to $ x $:
   \[
   AC'(x) = -\frac{32,400}{x^2} + 0.01
   \]
   Set the derivative equal to zero to find the critical points:
   \[
   -\frac{32,400}{x^2} + 0.01 = 0
   \]
   Solve for $ x $:
   \[
   \frac{32,400}{x^2} = 0.01
   \]
   \[
   x^2 = \frac{32,400}{0.01} = 3,240,000
   \]
   \[
   x = \sqrt{3,240,000} = 1,800
   \]

3. **Verify that this is a minimum:**
   To ensure that $ x = 1,800 $ minimizes the average cost, we check the second derivative of $ AC(x) $:
   \[
   AC''(x) = \frac{2 \times 32,400}{x^3}
   \]
   Since $ AC''(x) > 0 $ for $ x > 0 $, the function is concave upwards, confirming that $ x = 1,800 $ is a minimum.

4. **Calculate the minimum average cost:**
   Substitute $ x = 1,800 $ into the average cost function $ AC(x) $:
   \[
   AC(1,800) = \frac{32,400}{1,800} + 100 + 0.01 \times 1,800
   \]
   \[
   AC(1,800) = 18 + 100 + 18 = 136
   \]

### Conclusion:
- The company should manufacture **1,800 iPods** to minimize the average cost.
- The minimum average cost per iPod is **\$136**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the difference between total cost and average cost?
2. How can you interpret the marginal cost in this context?
3. What is the significance of the second derivative in optimization problems?
4. How would the cost change if the fixed cost increased?
5. How do economies of scale affect average costs in production?

**Tip:** In optimization problems, always check the second derivative to confirm whether you have found a maximum or minimum.

Assume that it costs Apple approximately C(x) = 32,400 + 100x + 0.01x^2 dollars to manufacture x 30-gigabyte video iPods in a day. How many iPods should be manufactured in order to minimize average cost? (Round to the nearest hundred.) What is the resulting average cost of an iPod? (Give your answer to the nearest dollar.)

Learn how to minimize the average cost of producing iPods using calculus. We derive the cost function C(x) = 32,400 + 100x + 0.01x^2 and use optimization techniques to find that producing 1,800 iPods minimizes the average cost, resulting in an average cost of $136 per iPod. Suitable for undergraduate students studying calculus and optimization.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation