The volume of a sphere is increasing at the rate of `8cm^(3)//s`. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.

Answer

To solve the problem step by step, we will use the formulas for the volume and surface area of a sphere and apply the chain rule of differentiation.

### Step 1: Write down the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Differentiate the volume with respect to time \( t \).
To find the rate at which the volume is changing with respect to time, we differentiate \( V \) with respect to \( t \):
\[ \frac{dV}{dt} = \frac{d}{dt} \left(\frac{4}{3} \pi r^3\right) = 4 \pi r^2 \frac{dr}{dt} \]

### Step 3: Substitute the known values.
We know that \( \frac{dV}{dt} = 8 \, \text{cm}^3/\text{s} \) and we need to find \( \frac{dr}{dt} \) when the radius \( r = 12 \, \text{cm} \).
Substituting these values into the differentiated volume equation:
\[ 8 = 4 \pi (12)^2 \frac{dr}{dt} \]

### Step 4: Solve for \( \frac{dr}{dt} \).
Calculating \( 4 \pi (12)^2 \):
\[ 4 \pi (12)^2 = 4 \pi \cdot 144 = 576 \pi \]
Now substituting this back into the equation:
\[ 8 = 576 \pi \frac{dr}{dt} \]
\[ \frac{dr}{dt} = \frac{8}{576 \pi} = \frac{1}{72 \pi} \, \text{cm/s} \]

### Step 5: Write down the formula for the surface area of a sphere.
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]

### Step 6: Differentiate the surface area with respect to time \( t \).
Differentiating \( A \) with respect to \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt} (4 \pi r^2) = 8 \pi r \frac{dr}{dt} \]

### Step 7: Substitute the known values to find \( \frac{dA}{dt} \).
Now, substituting \( r = 12 \, \text{cm} \) and \( \frac{dr}{dt} = \frac{1}{72 \pi} \):
\[ \frac{dA}{dt} = 8 \pi (12) \left(\frac{1}{72 \pi}\right) \]
\[ \frac{dA}{dt} = 8 \cdot 12 \cdot \frac{1}{72} = \frac{96}{72} = \frac{4}{3} \, \text{cm}^2/\text{s} \]

### Final Answer:
The rate at which the surface area is increasing when the radius of the sphere is 12 cm is:
\[ \frac{dA}{dt} = \frac{4}{3} \, \text{cm}^2/\text{s} \]