The ratio of the surface area of sphere A to the surface area of sphere B is 729 : 1. What is ratio of the volume of sphere A to sphere B ?

Options

A. `27 : 1`
B. `81 : 1`
C. `19,683 : 1`
D. `26, 224 : 1`

Explanation

To find the ratio of the volume of sphere A to the volume of sphere B given the ratio of their surface areas, we can follow these steps:

### Step 1: Understand the formulas
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]

### Step 2: Set up the ratio of surface areas
We are given that the ratio of the surface area of sphere A to the surface area of sphere B is:
\[ \frac{S_A}{S_B} = \frac{729}{1} \]
This can be expressed using the surface area formula:
\[ \frac{4\pi r_A^2}{4\pi r_B^2} = \frac{729}{1} \]
The \( 4\pi \) cancels out, simplifying to:
\[ \frac{r_A^2}{r_B^2} = 729 \]

### Step 3: Take the square root to find the ratio of radii
Taking the square root of both sides gives us the ratio of the radii:
\[ \frac{r_A}{r_B} = \sqrt{729} = 27 \]
Thus, the ratio of the radii \( r_A : r_B \) is:
\[ 27 : 1 \]

### Step 4: Set up the ratio of volumes
Now, we need to find the ratio of the volumes of spheres A and B:
\[ \frac{V_A}{V_B} = \frac{\frac{4}{3}\pi r_A^3}{\frac{4}{3}\pi r_B^3} \]
Again, the \( \frac{4}{3}\pi \) cancels out, simplifying to:
\[ \frac{V_A}{V_B} = \frac{r_A^3}{r_B^3} \]

### Step 5: Substitute the ratio of radii
Now we substitute the ratio of the radii we found earlier:
\[ \frac{V_A}{V_B} = \left(\frac{r_A}{r_B}\right)^3 = \left(27\right)^3 \]
Calculating \( 27^3 \):
\[ 27^3 = 19683 \]
Thus, the ratio of the volumes is:
\[ \frac{V_A}{V_B} = 19683 : 1 \]

### Final Answer
The ratio of the volume of sphere A to the volume of sphere B is:
\[ 19683 : 1 \]

---