To answer the question, we'll use the differential approximation to estimate the change in average cost per racket based on the given cost function.

The formula for the average cost $C(x)$ per racket is: $C(x) = \frac{370}{x} + 9 + \frac{1}{3}x$ where $x$ is the number of rackets produced per hour.

Step 1: Compute the derivative of $C(x)$

The derivative of $C(x)$ with respect to $x$, denoted as $C'(x)$, will give us the rate of change of the average cost. We differentiate each term of the function:

$C(x) = \frac{370}{x} + 9 + \frac{1}{3}x$

• The derivative of $\frac{370}{x}$ is $-\frac{370}{x^2}$.
• The derivative of $9$ is $0$ (since it's a constant).
• The derivative of $\frac{1}{3}x$ is $\frac{1}{3}$.

So, the derivative is: $C'(x) = -\frac{370}{x^2} + \frac{1}{3}$

Step 2: Approximate the change in average cost

The differential $dC$ gives us the approximate change in cost: $dC \approx C'(x) \cdot dx$ where $dx$ is the change in $x$ (the number of rackets produced per hour).

Part (A): Increase in production from 10 to 15 per hour

Here, we need to calculate the approximate change when $x$ changes from 10 to 15. This means:

• Initial value: $x = 10$
• Final value: $x = 15$
• $dx = 15 - 10 = 5$

First, we calculate $C'(10)$: $C'(10) = -\frac{370}{10^2} + \frac{1}{3} = -\frac{370}{100} + \frac{1}{3} = -3.7 + 0.3333 \approx -3.3667$

Now, approximate the change in cost: $dC \approx C'(10) \cdot dx = -3.3667 \cdot 5 = -16.8333$

So, the average cost per racket will decrease by approximately $16.83.

Part (B): Increase in production from 35 to 40 per hour

Here, we calculate the approximate change when $x$ changes from 35 to 40:

• Initial value: $x = 35$
• Final value: $x = 40$
• $dx = 40 - 35 = 5$

First, we calculate $C'(35)$: $C'(35) = -\frac{370}{35^2} + \frac{1}{3} = -\frac{370}{1225} + \frac{1}{3} \approx -0.3020 + 0.3333 = 0.0313$

Now, approximate the change in cost: $dC \approx C'(35) \cdot dx = 0.0313 \cdot 5 = 0.1565$

So, the average cost per racket will increase by approximately $0.16.

Final answers:

• (A) If production is increased from 10 per hour to 15 per hour, the average cost per racket will decrease by approximately $16.83.
• (B) If production is increased from 35 per hour to 40 per hour, the average cost per racket will increase by approximately $0.16.

---

Would you like further details on how I derived these results, or do you have other questions? Here are some related questions for further exploration:

1. How would the differential approximation change if we used a smaller or larger $dx$?
2. How does the rate of change of the cost function impact the overall cost at different production levels?
3. What does the second derivative of the cost function tell us about the nature of the cost curve?
4. How would we approach this problem if the cost function included additional fixed costs or variables?
5. How can we use the derivative of the cost function to identify the production level that minimizes cost?

Tip: The differential approximation works best for small changes in $x$, but larger changes can lead to less accurate estimates. Always check how small $dx$ is to ensure the approximation is valid.