The volume of cube is increasing at a rate of `9 cm^(3)//sec`. Find the rate of increase of its surface area when the side of the cube is 10 cm.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript while ensuring clarity and precision in our calculations.

### Step 1: Understand the given information
We know:
- The volume of the cube is increasing at a rate of \( \frac{dV}{dt} = 9 \, \text{cm}^3/\text{sec} \).
- The side length of the cube at the moment we are interested in is \( x = 10 \, \text{cm} \).

### Step 2: Write the formula for the volume of a cube
The volume \( V \) of a cube with side length \( x \) is given by:
\[ V = x^3 \]

### Step 3: Differentiate the volume with respect to time
To find how the volume changes with time, we differentiate \( V \) with respect to \( t \):
\[ \frac{dV}{dt} = 3x^2 \frac{dx}{dt} \]

### Step 4: Substitute the known values
We know \( \frac{dV}{dt} = 9 \, \text{cm}^3/\text{sec} \) and \( x = 10 \, \text{cm} \). Plugging these values into the differentiated equation:
\[ 9 = 3(10^2) \frac{dx}{dt} \]
\[ 9 = 3(100) \frac{dx}{dt} \]
\[ 9 = 300 \frac{dx}{dt} \]

### Step 5: Solve for \( \frac{dx}{dt} \)
Now, we can solve for \( \frac{dx}{dt} \):
\[ \frac{dx}{dt} = \frac{9}{300} = \frac{3}{100} \, \text{cm/sec} \]

### Step 6: Write the formula for the surface area of a cube
The surface area \( S \) of a cube is given by:
\[ S = 6x^2 \]

### Step 7: Differentiate the surface area with respect to time
To find the rate of change of the surface area, we differentiate \( S \) with respect to \( t \):
\[ \frac{dS}{dt} = 12x \frac{dx}{dt} \]

### Step 8: Substitute the known values
Now we substitute \( x = 10 \, \text{cm} \) and \( \frac{dx}{dt} = \frac{3}{100} \, \text{cm/sec} \):
\[ \frac{dS}{dt} = 12(10) \left(\frac{3}{100}\right) \]
\[ \frac{dS}{dt} = 120 \left(\frac{3}{100}\right) \]
\[ \frac{dS}{dt} = \frac{360}{100} = 3.6 \, \text{cm}^2/\text{sec} \]

### Final Answer
The rate of increase of the surface area when the side of the cube is 10 cm is:
\[ \frac{dS}{dt} = 3.6 \, \text{cm}^2/\text{sec} \]