The function *f*(*x*) = xπ−−√xπ gives the diameter, in inches, of a proposed spherical sculpture with a surface area of *x* square inches. The artist making the sculpture wants to know how the diameter changes if the surface area is increased. What is the average rate of change for the function as the surface area changes from 12.6 in.2 to 28.3 in.2? Round your answer to the nearest hundredth of an inch.

average rate of change =  in.

To calculate the average rate of change of a function \( f(x) \), we use the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Where:
- \( a \) and \( b \) are the two values of \( x \) (surface area in this case),
- \( f(x) \) represents the function for the diameter of the sculpture, \( f(x) = \sqrt{\pi x} \).

Given:
- \( a = 12.6 \, \text{in}^2 \)
- \( b = 28.3 \, \text{in}^2 \)

Let's calculate the average rate of change using the given values.

1. Compute \( f(12.6) = \sqrt{\pi \cdot 12.6} \)
2. Compute \( f(28.3) = \sqrt{\pi \cdot 28.3} \)
3. Apply the formula for the average rate of change.

I'll calculate this now.The average rate of change of the diameter as the surface area changes from 12.6 square inches to 28.3 square inches is approximately \( 0.20 \) inches per square inch, rounded to the nearest hundredth.

Would you like any further details or clarification?

Here are 5 related questions for further exploration:
1. How does the diameter change if the surface area doubles?
2. What is the surface area of the sculpture if the diameter is known?
3. How can you use calculus to find the instantaneous rate of change for this function?
4. How does the surface area formula for a sphere relate to this problem?
5. What would the rate of change be if the surface area increased from 28.3 to 50 in²?

**Tip:** The average rate of change provides insight into how a quantity changes over an interval, but the instantaneous rate of change (derivative) tells you how it changes at a specific point.

The function f(x) = √(πx) gives the diameter, in inches, of a proposed spherical sculpture with a surface area of x square inches. The artist making the sculpture wants to know how the diameter changes if the surface area is increased. What is the average rate of change for the function as the surface area changes from 12.6 in² to 28.3 in²? Round your answer to the nearest hundredth of an inch.

Learn how to calculate the average rate of change of a spherical sculpture's diameter as the surface area increases from 12.6 in² to 28.3 in². This problem involves the function f(x) = √(πx), key formulas for average rate of change, and geometric concepts related to spheres. Suitable for high school students in Grades 9-12.

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