The volumes of spheres A and B are in the ratio `125 : 64`. If the sum of radi of A and B is 36 cm, then the surface area (in `cm^(2)`) of A is :

Options

A. 1800`pi`
B. 512`pi`
C. 1600`pi`
D. 1024`pi`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the Volume Ratio
The volumes of spheres A and B are given in the ratio \(125 : 64\). We can express this ratio in terms of the radii of the spheres.

### Step 2: Use the Volume Formula
The volume \(V\) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Let the radius of sphere A be \(r_A\) and the radius of sphere B be \(r_B\). According to the problem, we have:
\[ \frac{V_A}{V_B} = \frac{125}{64} \]
Substituting the volume formula:
\[ \frac{\frac{4}{3} \pi r_A^3}{\frac{4}{3} \pi r_B^3} = \frac{125}{64} \]
The \(\frac{4}{3} \pi\) cancels out:
\[ \frac{r_A^3}{r_B^3} = \frac{125}{64} \]

### Step 3: Relate Radii Using Cube Roots
We can express the ratio of the radii by taking the cube root:
\[ \frac{r_A}{r_B} = \frac{5}{4} \]
Let \(r_A = 5k\) and \(r_B = 4k\) for some constant \(k\).

### Step 4: Use the Sum of Radii
According to the problem, the sum of the radii is:
\[ r_A + r_B = 36 \text{ cm} \]
Substituting the expressions for \(r_A\) and \(r_B\):
\[ 5k + 4k = 36 \]
This simplifies to:
\[ 9k = 36 \]

### Step 5: Solve for \(k\)
Dividing both sides by 9:
\[ k = 4 \]

### Step 6: Find the Radii
Now we can find the radii:
\[ r_A = 5k = 5 \times 4 = 20 \text{ cm} \]
\[ r_B = 4k = 4 \times 4 = 16 \text{ cm} \]

### Step 7: Calculate the Surface Area of Sphere A
The surface area \(S\) of a sphere is given by the formula:
\[ S = 4 \pi r^2 \]
Substituting \(r_A\):
\[ S_A = 4 \pi (20)^2 = 4 \pi \times 400 = 1600 \pi \text{ cm}^2 \]

### Final Answer
The surface area of sphere A is:
\[ \boxed{1600 \pi \text{ cm}^2} \]