If the radii of two sphere are in the ratio `1:4`, then their surface area are in the ratio :

Options

A. `1:2`
B. `1:4`
C. `1:8`
D. `1:16`

Answer

To solve the problem of finding the ratio of the surface areas of two spheres given that their radii are in the ratio of 1:4, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Ratio of Radii**:
- Let the radius of the first sphere be \( r_1 = 1 \) (unit).
- Let the radius of the second sphere be \( r_2 = 4 \) (units).
- Therefore, the ratio of the radii is \( r_1 : r_2 = 1 : 4 \).

2. **Formula for Surface Area of a Sphere**:
- The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]

3. **Calculate the Surface Area of the First Sphere**:
- Using the radius \( r_1 = 1 \):
\[ A_1 = 4 \pi (1^2) = 4 \pi \]

4. **Calculate the Surface Area of the Second Sphere**:
- Using the radius \( r_2 = 4 \):
\[ A_2 = 4 \pi (4^2) = 4 \pi (16) = 64 \pi \]

5. **Find the Ratio of the Surface Areas**:
- Now, we can find the ratio of the surface areas \( A_1 : A_2 \):
\[ A_1 : A_2 = 4 \pi : 64 \pi \]
- The \( \pi \) cancels out:
\[ A_1 : A_2 = 4 : 64 \]
- Simplifying this ratio:
\[ A_1 : A_2 = 1 : 16 \]

### Final Answer:
The ratio of the surface areas of the two spheres is \( 1 : 16 \).

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