The volume of two spheres are in the ratio 216 : 125. The difference of their surface areas, if the sum of their radii is 11 units, is

Options

A. `38 pi` sq. units
B. `45 pi` sq. units
C. `50 pi` sq. units
D. `44 pi` sq. units

Answer

To solve the problem step by step, let's follow the given information and derive the required values.

### Step 1: Understand the Volume Ratio
The volumes of two spheres are given in the ratio 216:125.

Let the volumes of the spheres be \( V_1 \) and \( V_2 \). We can express this as:
\[ \frac{V_1}{V_2} = \frac{216}{125} \]

### Step 2: Relate Volume to Radius
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, we can write:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi R_1^3}{\frac{4}{3} \pi R_2^3} = \frac{R_1^3}{R_2^3} \]

### Step 3: Set Up the Equation
From the volume ratio, we can equate:
\[ \frac{R_1^3}{R_2^3} = \frac{216}{125} \]

### Step 4: Take the Cube Root
Taking the cube root of both sides gives us the ratio of the radii:
\[ \frac{R_1}{R_2} = \frac{\sqrt[3]{216}}{\sqrt[3]{125}} = \frac{6}{5} \]

### Step 5: Express Radii in Terms of a Variable
Let \( R_1 = 6k \) and \( R_2 = 5k \) for some variable \( k \).

### Step 6: Use the Sum of Radii
We know that the sum of the radii is given as:
\[ R_1 + R_2 = 11 \]
Substituting the expressions for \( R_1 \) and \( R_2 \):
\[ 6k + 5k = 11 \implies 11k = 11 \implies k = 1 \]

### Step 7: Calculate the Radii
Now substituting back for \( k \):
\[ R_1 = 6 \times 1 = 6 \quad \text{and} \quad R_2 = 5 \times 1 = 5 \]

### Step 8: Calculate the Surface Areas
The surface area \( S \) of a sphere is given by:
\[ S = 4 \pi r^2 \]
Calculating the surface areas for both spheres:
\[ S_1 = 4 \pi R_1^2 = 4 \pi (6^2) = 4 \pi (36) = 144 \pi \]
\[ S_2 = 4 \pi R_2^2 = 4 \pi (5^2) = 4 \pi (25) = 100 \pi \]

### Step 9: Calculate the Difference in Surface Areas
Now we find the difference between the surface areas:
\[ \text{Difference} = S_1 - S_2 = 144 \pi - 100 \pi = 44 \pi \]

### Final Answer
The difference in their surface areas is:
\[ \boxed{44 \pi} \text{ square units} \]