The ratio between the volumes of two spheres is 8 : 27. What is the ratiobetween their surface areas?

Options

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

Explanation

To find the ratio between 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 ratio of volumes
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: Express the volumes in terms of their radii
Let the radii of the two spheres be \( r_1 \) and \( r_2 \). Then we can express the volumes as:
\[ V_1 = \frac{4}{3} \pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi r_2^3 \]
Thus, the ratio of the volumes becomes:
\[ \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{r_1^3}{r_2^3} \]

### Step 4: Set up the equation
From the volume ratio, we have:
\[ \frac{r_1^3}{r_2^3} = \frac{8}{27} \]

### Step 5: Take the cube root to find the ratio of the radii
Taking the cube root of both sides gives us:
\[ \frac{r_1}{r_2} = \frac{2}{3} \]

### Step 6: Find 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 surface areas of the two spheres are:
\[ A_1 = 4 \pi r_1^2 \quad \text{and} \quad A_2 = 4 \pi r_2^2 \]
The ratio of the surface areas is:
\[ \frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{r_1^2}{r_2^2} \]

### Step 7: Substitute the ratio of the radii
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
Thus, the ratio between the surface areas of the two spheres is:
\[ \frac{A_1}{A_2} = \frac{4}{9} \]

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