The radius of a balloon is increasing at the rate of 10 cm/sec. At what rate is the surface area of the balloon increasing when the radius is 15 cm?

Explanation

To solve the problem, we need to find the rate at which the surface area of a balloon is increasing when the radius is 15 cm, given that the radius is increasing at a rate of 10 cm/sec.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a balloon in the shape of a sphere. The radius \( r \) of the balloon is increasing at a rate of \( \frac{dr}{dt} = 10 \) cm/sec. We need to find the rate of change of the surface area \( S \) when the radius is \( r = 15 \) cm.

2. **Formula for Surface Area**:
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