The radius of spherical bubble is changing with time. The rate of change of its volume is given by :

Options

A. `4pir^(2)(dr)/(dt)`
B. `(4)/(3)pir^(2)`
C. `(8)/(3)pir^(2)`
D. `(8)/(3)pir(dr)/(dt)`

Answer

To solve the problem regarding the rate of change of the volume of a spherical bubble as its radius changes with time, we can follow these steps:

### Step 1: Write down the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
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 volume with respect to time, we need to differentiate \( V \) with respect to \( t \):
\[ \frac{dV}{dt} = \frac{d}{dt}\left(\frac{4}{3} \pi r^3\right) \]

### Step 3: Apply the chain rule.
Using the chain rule for differentiation, we have:
\[ \frac{dV}{dt} = \frac{4}{3} \pi \cdot \frac{d}{dt}(r^3) = \frac{4}{3} \pi \cdot 3r^2 \cdot \frac{dr}{dt} \]
Here, \( \frac{dr}{dt} \) is the rate of change of the radius with respect to time.

### Step 4: Simplify the expression.
The \( 3 \) in the numerator and denominator cancels out:
\[ \frac{dV}{dt} = 4 \pi r^2 \cdot \frac{dr}{dt} \]

### Step 5: Write the final result.
Thus, the rate of change of the volume of the spherical bubble is given by:
\[ \frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt} \]

### Conclusion:
The correct answer is \( 4 \pi r^2 \frac{dr}{dt} \).

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