https://file.aiquickdraw.com/mathbot/file/1726655173834raa9eysw.jpg

To determine the rate at which the average profit per belt is changing, we need to follow the steps below:

### Step 1: Profit and Average Profit Formulas
The profit \( P(x) \) is given by the difference between revenue \( R(x) \) and cost \( C(x) \):
\[
P(x) = R(x) - C(x)
\]

Thus, the **average profit** per belt is:
\[
\overline{P}(x) = \frac{P(x)}{x} = \frac{R(x) - C(x)}{x}
\]

### Step 2: Revenue and Cost Functions
From the image, we have the following:
- Cost function: \( C(x) = 710 + 37x - 0.062x^2 \)
- Revenue function: \( R(x) = 55x^{\frac{9}{10}} \)

Now, we substitute these into the formula for \( P(x) \):
\[
P(x) = 55x^{\frac{9}{10}} - (710 + 37x - 0.062x^2)
\]

Simplifying this:
\[
P(x) = 55x^{\frac{9}{10}} - 710 - 37x + 0.062x^2
\]

### Step 3: Average Profit Formula
Now, the average profit per belt is:
\[
\overline{P}(x) = \frac{55x^{\frac{9}{10}} - 710 - 37x + 0.062x^2}{x}
\]

This simplifies to:
\[
\overline{P}(x) = \frac{55x^{\frac{9}{10}}}{x} - \frac{710}{x} - 37 + 0.062x
\]
\[
\overline{P}(x) = 55x^{-\frac{1}{10}} - \frac{710}{x} - 37 + 0.062x
\]

### Step 4: Rate of Change of Average Profit
To find the rate of change of the average profit, we need to differentiate \( \overline{P}(x) \) with respect to \( x \):
\[
\overline{P}'(x) = \frac{d}{dx}\left(55x^{-\frac{1}{10}} - \frac{710}{x} - 37 + 0.062x\right)
\]

Let's compute this derivative.

The derivative of the average profit function \( \overline{P}(x) \), which represents the rate at which the average profit is changing, is:

\[
\overline{P}'(x) = 710x^{-2} - 5.5x^{-1.1} + 0.062
\]

### Step 5: Evaluate at \( x = 175 \)
Now, we substitute \( x = 175 \) to find the rate at which the average profit is changing when 175 belts have been produced and sold.

Let's calculate that.The rate at which the average profit is changing when 175 belts have been produced is approximately \( 0.0664 \) dollars per belt.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What does the sign of \( \overline{P}'(x) \) indicate about profit at \( x = 175 \)?
2. How does the cost function affect the average profit for different production levels?
3. What is the significance of \( \frac{R(x) - C(x)}{x} \) in terms of maximizing average profit?
4. How does the average profit compare to marginal profit in this context?
5. How would increasing the efficiency of production impact the cost function \( C(x) \)?

**Tip:** Understanding the behavior of a function's derivative is essential in optimizing business decisions like production rates for maximizing profit.

An accessories company finds that the cost and revenue, in dollars, of producing x belts is given by C(x) = 710 + 37x - 0.062x^2 and R(x) = 55x^(9/10), respectively. Determine the rate at which the accessories company's average profit per belt is changing when 175 belts have been produced and sold. First, find the rate at which the average profit is changing when x belts have been produced.

Learn how to calculate the rate at which the average profit per belt is changing for an accessories company, given the cost and revenue functions. This problem involves applying concepts of differentiation and understanding the behavior of revenue and cost functions. Suitable for students in Grades 10-12.

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