The volume of a spherical balloon is increasing at a rate of ` 25 cm^(3)//sec`. Find the rate of increase of its curved surface when the radius of balloon is 5 cm.

Answer

To solve the problem, we need to find the rate of increase of the curved surface area of a spherical balloon when its radius is 5 cm, given that the volume is increasing at a rate of 25 cm³/sec.

### Step 1: Write down the formulas for volume and surface area of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
The curved surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]

### Step 2: Differentiate the volume with respect to time.
We need to 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 get:
\[ \frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt} \]

### Step 3: Substitute the known values.
We know that \( \frac{dV}{dt} = 25 \, \text{cm}^3/\text{sec} \) and \( r = 5 \, \text{cm} \). Substitute these values into the equation:
\[ 25 = 4 \pi (5^2) \frac{dr}{dt} \]
Calculating \( 5^2 \):
\[ 25 = 4 \pi (25) \frac{dr}{dt} \]
Simplifying:
\[ 25 = 100 \pi \frac{dr}{dt} \]

### Step 4: Solve for \( \frac{dr}{dt} \).
Rearranging the equation gives:
\[ \frac{dr}{dt} = \frac{25}{100 \pi} = \frac{1}{4 \pi} \, \text{cm/sec} \]

### Step 5: Differentiate the surface area with respect to time.
Now, we differentiate the surface area with respect to time:
\[ \frac{dA}{dt} = \frac{d}{dt} (4 \pi r^2) = 8 \pi r \frac{dr}{dt} \]

### Step 6: Substitute \( r \) and \( \frac{dr}{dt} \) into the surface area derivative.
Substituting \( r = 5 \, \text{cm} \) and \( \frac{dr}{dt} = \frac{1}{4 \pi} \):
\[ \frac{dA}{dt} = 8 \pi (5) \left( \frac{1}{4 \pi} \right) \]
Simplifying:
\[ \frac{dA}{dt} = 8 \cdot 5 \cdot \frac{1}{4} = 10 \, \text{cm}^2/\text{sec} \]

### Final Result:
The rate of increase of the curved surface area of the balloon when the radius is 5 cm is:
\[ \frac{dA}{dt} = 10 \, \text{cm}^2/\text{sec} \]

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