If the ratio of volumes of two spheres is 1 : 8, then the ratio of their surface areas is

Options

A. `1 : 2`
B. `1 : 4`
C. `1 : 6`
D. `1 : 8`

Answer

To solve the problem of finding the ratio of the surface areas of two spheres given the ratio of their volumes, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Ratio**:
We are given that the ratio of the volumes of two spheres is 1:8. This can be expressed mathematically as:
\[ \frac{V_1}{V_2} = \frac{1}{8} \]

2. **Volume Formula for Spheres**:
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.

3. **Set Up the Volume Equation**:
Let the radius of the first sphere be \(R_1\) and the radius of the second sphere be \(R_2\). Thus, we can write:
\[ \frac{\frac{4}{3} \pi R_1^3}{\frac{4}{3} \pi R_2^3} = \frac{1}{8} \]
The \(\frac{4}{3} \pi\) cancels out from both sides, leading to:
\[ \frac{R_1^3}{R_2^3} = \frac{1}{8} \]

4. **Relate the Radii**:
From the above equation, we can deduce:
\[ R_1^3 : R_2^3 = 1 : 8 \]
Taking the cube root of both sides gives us:
\[ \frac{R_1}{R_2} = \frac{1}{2} \]
This means that \(R_1\) is half of \(R_2\) or \(R_2 = 2R_1\).

5. **Surface Area Formula for Spheres**:
The surface area \(A\) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
Therefore, the surface areas of the two spheres can be expressed as:
\[ A_1 = 4 \pi R_1^2 \quad \text{and} \quad A_2 = 4 \pi R_2^2 \]

6. **Set Up the Surface Area Ratio**:
The ratio of the surface areas is:
\[ \frac{A_1}{A_2} = \frac{4 \pi R_1^2}{4 \pi R_2^2} \]
The \(4 \pi\) cancels out, leading to:
\[ \frac{R_1^2}{R_2^2} \]

7. **Substitute the Radius Ratio**:
We already found that \(\frac{R_1}{R_2} = \frac{1}{2}\). Therefore:
\[ \frac{R_1^2}{R_2^2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

8. **Conclusion**:
The ratio of the surface areas of the two spheres is:
\[ \frac{A_1}{A_2} = \frac{1}{4} \]
This can also be expressed as:
\[ 1 : 4 \]

### Final Answer:
The ratio of the surface areas of the two spheres is \(1 : 4\).