The volume of two sphere are in the ratio 8:27. The ratio of their surface area is :

Options

A. `4:9`
B. `2:3`
C. `4:5`
D. `5:6`

Answer

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

### Step 1: Understand the relationship between volume and radius
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: Set up the volume ratio
Let the volumes of the two spheres be \( V_1 \) and \( V_2 \). According to the problem, the ratio of their volumes is:
\[ \frac{V_1}{V_2} = \frac{8}{27} \]

### Step 3: Relate the volume ratio to the radius ratio
From the volume formula, we can express the ratio of the volumes in terms of the radii:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{r_1^3}{r_2^3} \]
Thus, we have:
\[ \frac{r_1^3}{r_2^3} = \frac{8}{27} \]

### Step 4: Find the ratio of the radii
To find the ratio of the radii \( \frac{r_1}{r_2} \), we take the cube root of both sides:
\[ \frac{r_1}{r_2} = \sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \]

### Step 5: Calculate the ratio of the surface areas
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
Thus, the ratio of the surface areas \( A_1 \) and \( A_2 \) can be expressed as:
\[ \frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{r_1^2}{r_2^2} \]
Now, substituting the ratio of the radii:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

### Final Answer
The ratio of the surface areas of the two spheres is:
\[ \frac{A_1}{A_2} = \frac{4}{9} \]

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