The sum of radii of two spheres is 10 cm and the sum of their volume is `880 cm^(3)`. What will be the product of their radii ?

Options

A. 21
B. `26(1)/(3)`
C. `33(1)/(3)`
D. 70

Answer

To solve the problem step by step, we will use the given information about the radii and volumes of the two spheres.

### Step 1: Define the variables
Let the radii of the two spheres be \( r_1 \) and \( r_2 \).

### Step 2: Set up the equations
From the problem, we have two equations based on the information provided:
1. The sum of the radii:
\[ r_1 + r_2 = 10 \quad \text{(1)} \]
2. The sum of the volumes:
\[ V_1 + V_2 = 880 \quad \text{(2)} \]
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Therefore, the volumes of the two spheres can be expressed as:
\[ V_1 = \frac{4}{3} \pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi r_2^3 \]
Substituting these into equation (2):
\[ \frac{4}{3} \pi r_1^3 + \frac{4}{3} \pi r_2^3 = 880 \]

### Step 3: Simplify the volume equation
We can factor out \( \frac{4}{3} \pi \) from the left side:
\[ \frac{4}{3} \pi (r_1^3 + r_2^3) = 880 \]
Now, we can solve for \( r_1^3 + r_2^3 \):
\[ r_1^3 + r_2^3 = \frac{880 \times 3}{4 \pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r_1^3 + r_2^3 = \frac{880 \times 3 \times 7}{4 \times 22} = \frac{18480}{88} = 210 \]

### Step 4: Use the identity for cubes
We can use the identity:
\[ r_1^3 + r_2^3 = (r_1 + r_2)(r_1^2 - r_1 r_2 + r_2^2) \]
We know \( r_1 + r_2 = 10 \), so:
\[ 210 = 10 (r_1^2 - r_1 r_2 + r_2^2) \]
Dividing both sides by 10:
\[ 21 = r_1^2 - r_1 r_2 + r_2^2 \quad \text{(3)} \]

### Step 5: Express \( r_1^2 + r_2^2 \)
Using the square of the sum of the radii:
\[ (r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2 \]
Substituting \( r_1 + r_2 = 10 \):
\[ 100 = r_1^2 + r_2^2 + 2r_1 r_2 \]
From equation (3), we can express \( r_1^2 + r_2^2 \):
\[ r_1^2 + r_2^2 = 21 + r_1 r_2 \]
Substituting this into the equation:
\[ 100 = 21 + r_1 r_2 + 2r_1 r_2 \]
This simplifies to:
\[ 100 = 21 + 3r_1 r_2 \]
\[ 79 = 3r_1 r_2 \]
\[ r_1 r_2 = \frac{79}{3} \]

### Conclusion
The product of the radii \( r_1 \) and \( r_2 \) is:
\[ \boxed{\frac{79}{3} \text{ cm}^2} \]