A company finds that the cost, in dollars, of producing handbags is given by
. Find the rate at which average cost, , is changing when
200 handbags have been produced. (Round to the nearest cent.)
The average cost is ? by
handbags are produced.

Ask by West Cross. in the United States

Jan 25,2025

Question

A company finds that the cost, in dollars, of producing handbags is given by
. Find the rate at which average cost, , is changing when
200 handbags have been produced. (Round to the nearest cent.)
The average cost is ? by
handbags are produced.

Ask by West Cross. in the United States

Jan 25,2025

UpStudy AI · Accepted Answer

To determine the rate at which the average cost is changing when 200 handbags have been produced, follow these steps:

1. **Cost Function**:
   $$
   C(t) = 465 + 25t + 10t^2
   $$
   where $$ t $$ is the number of handbags produced.

2. **Average Cost Function**:
   $$
   \text{Average Cost} = \frac{C(t)}{t} = \frac{465}{t} + 25 + 10t
   $$

3. **Derivative of the Average Cost**:
   To find how the average cost is changing with respect to $$ t $$, take the derivative of the average cost function:
   $$
   \frac{d}{dt}\left(\frac{C(t)}{t}\right) = \frac{d}{dt}\left(\frac{465}{t} + 25 + 10t\right) = -\frac{465}{t^2} + 10
   $$

4. **Evaluate at $$ t = 200 $$**:
   $$
   \frac{d}{dt}\left(\frac{C(t)}{t}\right)\Bigg|_{t=200} = -\frac{465}{200^2} + 10 = -\frac{465}{40000} + 10 \approx -0.011625 + 10 = 9.988375
   $$
   Rounded to the nearest cent:
   $$
   \approx \$9.99
   $$

**Conclusion**:
When 200 handbags have been produced, the average cost is **increasing by \$9.99** per additional handbag produced.

Answer: 
When 200 handbags have been produced, the average cost is increasing by $ 9.99 per handbag.
When 200 handbags are produced, the average cost is increasing by $9.99 per handbag.

A company finds that the cost, in dollars, of producing handbags is given by
. Find the rate at which average cost, , is changing when
200 handbags have been produced. (Round to the nearest cent.)
The average cost is ? by
handbags are produced.

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A company finds that the cost, in dollars, of producing handbags is given by . Find the rate at which average cost, , is changing when 200 handbags have been produced. (Round to the nearest cent.) The average cost is ? by handbags are produced.

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A company finds that the cost, in dollars, of producing handbags is given by
. Find the rate at which average cost, , is changing when
200 handbags have been produced. (Round to the nearest cent.)
The average cost is ? by
handbags are produced.