The volume of a sphere is increasing at the rate of `4pi` `cm^3//sec` . The rate of increase of the radius when the volume is `288pi` `cm^3` , is

Options

A. (a) `1/6`
B. (b) `1/9`
C. (c) `1/36`
D. (d) `1/24`

Explanation

To solve the problem step by step, we will follow these procedures:

### Step 1: Understand the relationship between volume and radius
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Differentiate the volume with respect to time
To find the rate of change of the radius with respect to time, we first differentiate the volume with respect to time \( t \):
\[ \frac{dV}{dt} = \frac{d}{dt} \left( \frac{4}{3} \pi r^3 \right) \]
Using the chain rule, we have:
\[ \frac{dV}{dt} = \frac{4}{3} \pi \cdot 3r^2 \frac{dr}{dt} = 4 \pi r^2 \frac{dr}{dt} \]

### Step 3: Substitute the known values
We know from the problem that \( \frac{dV}{dt} = 4\pi \, \text{cm}^3/\text{sec} \) and we need to find \( \frac{dr}{dt} \) when the volume \( V = 288\pi \, \text{cm}^3 \).

### Step 4: Find the radius when the volume is \( 288\pi \)
To find the radius \( r \) when the volume is \( 288\pi \), we set up the equation:
\[ 288\pi = \frac{4}{3} \pi r^3 \]
Dividing both sides by \( \pi \):
\[ 288 = \frac{4}{3} r^3 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ r^3 = 288 \cdot \frac{3}{4} = 216 \]
Taking the cube root of both sides:
\[ r = \sqrt[3]{216} = 6 \, \text{cm} \]

### Step 5: Substitute \( r \) back into the differentiated equation
Now we substitute \( r = 6 \, \text{cm} \) into the differentiated equation:
\[ 4\pi = 4\pi (6^2) \frac{dr}{dt} \]
This simplifies to:
\[ 4\pi = 4\pi \cdot 36 \frac{dr}{dt} \]
Dividing both sides by \( 4\pi \):
\[ 1 = 36 \frac{dr}{dt} \]

### Step 6: Solve for \( \frac{dr}{dt} \)
Now, we solve for \( \frac{dr}{dt} \):
\[ \frac{dr}{dt} = \frac{1}{36} \, \text{cm/sec} \]

### Final Answer
The rate of increase of the radius when the volume is \( 288\pi \, \text{cm}^3 \) is:
\[ \frac{dr}{dt} = \frac{1}{36} \, \text{cm/sec} \]

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