https://file.aiquickdraw.com/mathbot/file/17260135832404tq6j2hh.jpg

I have opened the image. Let's focus on solving the problem as specified:

The given cost function is:
\[
\overline{C}(x) = 0.6x^2 + 19x - 262 + \frac{2700}{x}
\]
where \( \overline{C}(x) \) represents the average cost per hour in dollars for producing \( x \) riding lawn mowers.

We need to solve two parts using a graphing utility:

1. **Determine the number of riding lawn mowers that minimizes the average cost**.
2. **Find the minimum average cost**.

Let me handle these steps using a graphing method for optimization.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can use any graphing utility like Desmos or a graphing calculator to plot the function and determine the minimum value visually. Here's how you can proceed:

1. **Plot the function** \[
   \overline{C}(x) = 0.6x^2 + 19x - 262 + \frac{2700}{x}
   \] 
   on a graphing tool.

2. Look for the value of \(x\) where the function reaches its lowest point. This value will be the number of lawn mowers that minimize the average cost.

If you need any help setting up the graph or interpreting it, let me know!

Do you have any questions? Here are five follow-up questions to expand your understanding:
1. What is the general method for finding the minimum or maximum of a function algebraically?
2. How can you verify the minimum cost using calculus?
3. What does the term "average cost" mean in this context?
4. How does adding the term \(\frac{2700}{x}\) affect the cost function?
5. Why is it important to round to the nearest whole number for practical purposes?

**Tip:** To quickly estimate the minimum value, set the derivative of the cost function to zero and solve for \(x\).

The average cost per hour in dollars of producing x riding lawn mowers is given by the following: C(x) = 0.6x^2 + 19x - 262 + 2700/x. (a) Use a graphing utility to determine the number of riding lawn mowers to produce in order to minimize average cost. (b) What is the minimum average cost?

Learn how to minimize the average cost per hour for producing riding lawn mowers by solving the function C(x) = 0.6x^2 + 19x - 262 + 2700/x. This problem involves quadratic and rational function optimization using graphing utilities. Suitable for students in Grades 10-12.

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