The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.

Explanation

To solve the problem step by step, we will use the concept of related rates in calculus.

### Step 1: Understand the problem
We have a hollow sphere with an inner radius \( r \) and an outer radius \( R \). The volume of the metal in the hollow sphere is constant. We know that the inner radius is increasing at a rate of \( \frac{dr}{dt} = 1 \) cm/sec. We need to find the rate of increase of the outer radius \( R \) when \( r = 4 \) cm and \( R = 8 \) cm.

### Step 2: Write the formula for the volume of the hollow sphere
The volume \( V \) of the hollow sphere is given by the formula:
\[
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