For a sphere of radius 10 cms. The numerical value of the surface area is how many percent of the numerical value of its volume ?

Options

A. `24%`
B. `26.5%`
C. `30%`
D. `45%`

Answer

To solve the problem of finding how many percent the surface area of a sphere is compared to its volume, we can follow these steps:

### Step 1: Calculate the Surface Area of the Sphere
The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Substituting the radius \( r = 10 \) cm into the formula:
\[ A = 4 \pi (10)^2 = 4 \pi (100) = 400 \pi \text{ cm}^2 \]

### Step 2: Calculate the Volume of the Sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the radius \( r = 10 \) cm into the formula:
\[ V = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi (1000) = \frac{4000}{3} \pi \text{ cm}^3 \]

### Step 3: Find the Percentage of Surface Area to Volume
To find out how many percent the surface area is of the volume, we use the formula:
\[ \text{Percentage} = \left( \frac{\text{Surface Area}}{\text{Volume}} \right) \times 100 \]
Substituting the values we calculated:
\[ \text{Percentage} = \left( \frac{400 \pi}{\frac{4000}{3} \pi} \right) \times 100 \]
The \( \pi \) cancels out:
\[ = \left( \frac{400}{\frac{4000}{3}} \right) \times 100 = \left( \frac{400 \times 3}{4000} \right) \times 100 \]
Simplifying this:
\[ = \left( \frac{1200}{4000} \right) \times 100 = \left( \frac{3}{10} \right) \times 100 = 30\% \]

### Conclusion
Thus, the numerical value of the surface area is **30%** of the numerical value of its volume.

---