The radius of a sphere is increasing at a rate of \[7.5\] meters per minute.

At a certain instant, the radius is \[5\] meters.

**What is the rate of change of the surface area of the sphere at that instant (in square meters per minute)?**

To find the rate of change of the surface area of the sphere with respect to time, we can use related rates. Let's work through the problem step by step.

### Step 1: Define the variables
- Let \( r(t) \) be the radius of the sphere as a function of time \( t \).
- Let \( A(t) \) be the surface area of the sphere as a function of time \( t \).

Given:
- The radius is increasing at a rate of \( \frac{dr}{dt} = 7.5 \) meters per minute.
- At the instant of interest, the radius \( r = 5 \) meters.

### Step 2: Write the formula for the surface area of the sphere
The surface area \( A \) of a sphere is given by the formula:
\[
A = 4\pi r^2
\]

### Step 3: Differentiate the surface area with respect to time \( t \)
To find the rate of change of the surface area \( \frac{dA}{dt} \), differentiate both sides of the equation with respect to \( t \):
\[
\frac{dA}{dt} = 4\pi \cdot 2r \cdot \frac{dr}{dt}
\]

This simplifies to:
\[
\frac{dA}{dt} = 8\pi r \cdot \frac{dr}{dt}
\]

### Step 4: Plug in the known values
Substitute the given values \( r = 5 \) meters and \( \frac{dr}{dt} = 7.5 \) meters per minute into the equation:
\[
\frac{dA}{dt} = 8\pi \times 5 \times 7.5
\]
\[
\frac{dA}{dt} = 300\pi \text{ square meters per minute}
\]

### Final Answer
The rate of change of the surface area of the sphere at that instant is \( 300\pi \) square meters per minute.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the rate of change of the surface area differ if the radius were increasing at a different rate?
2. What is the rate of change of the volume of the sphere at the same instant?
3. How does the rate of change of the surface area depend on the radius?
4. If the radius were decreasing instead of increasing, how would that affect the rate of change of the surface area?
5. Can you derive the rate of change of surface area for a sphere with a radius increasing exponentially?

**Tip:** For any related rates problem, always identify the key formula relating the variables and differentiate with respect to time to find the rate of change.

Discover how to calculate the rate of change of a sphere's surface area when its radius is increasing at a given rate. This problem involves related rates in calculus, focusing on the surface area formula A = 4\pi r^2. Suitable for high school students and calculus learners.

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