The surface area of two spheres are in the ratio `16:9` .Find the ratio of their volumes.

Answer

To find the ratio of the volumes of two spheres given that their surface areas are in the ratio of 16:9, we can follow these steps:

### Step 1: Understand the formula for surface area of a sphere
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Set up the ratio of the surface areas
Let the surface areas of the two spheres be \( S_1 \) and \( S_2 \), and their respective radii be \( r_1 \) and \( r_2 \). According to the problem, we have:
\[ \frac{S_1}{S_2} = \frac{16}{9} \]
This can be expressed using the formula for surface area:
\[ \frac{4\pi r_1^2}{4\pi r_2^2} = \frac{16}{9} \]
The \( 4\pi \) cancels out, simplifying to:
\[ \frac{r_1^2}{r_2^2} = \frac{16}{9} \]

### Step 3: Take the square root to find the ratio of the radii
Taking the square root of both sides gives:
\[ \frac{r_1}{r_2} = \frac{4}{3} \]

### Step 4: Use the formula for volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]
Let the volumes of the two spheres be \( V_1 \) and \( V_2 \):
\[ V_1 = \frac{4}{3}\pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3}\pi r_2^3 \]

### Step 5: Set up the ratio of the volumes
Now, we can find the ratio of the volumes:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} \]
Again, the \( \frac{4}{3}\pi \) cancels out:
\[ \frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} \]

### Step 6: Substitute the ratio of the radii
We already found that:
\[ \frac{r_1}{r_2} = \frac{4}{3} \]
Now, we cube this ratio:
\[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 = \left(\frac{4}{3}\right)^3 = \frac{64}{27} \]

### Final Answer
Thus, the ratio of the volumes of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{64}{27} \]

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