The sum of the volume of two solid spheres is `1144/3cm^(3)`. If the sum of their radii is 7 cm, then what will be the difference of the radi

Options

A. 4 cm
B. 3 cm
C. 2 cm
D. 1 cm

Answer

To solve the problem step-by-step, we will follow the mathematical reasoning outlined in the video transcript.

### Step 1: Define Variables
Let the radii of the two spheres be \( r_1 \) and \( r_2 \). We know from the problem that:
- The sum of the radii is given by:
\[ r_1 + r_2 = 7 \, \text{cm} \]

### Step 2: Volume of Spheres
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the volumes of the two spheres can be expressed as:
- Volume of sphere 1:
\[ V_1 = \frac{4}{3} \pi r_1^3 \]
- Volume of sphere 2:
\[ V_2 = \frac{4}{3} \pi r_2^3 \]

### Step 3: Sum of Volumes
According to the problem, the sum of the volumes is:
\[ V_1 + V_2 = \frac{1144}{3} \, \text{cm}^3 \]
Substituting the expressions for \( V_1 \) and \( V_2 \):
\[ \frac{4}{3} \pi r_1^3 + \frac{4}{3} \pi r_2^3 = \frac{1144}{3} \]

### Step 4: Simplify the Volume Equation
Factor out \( \frac{4}{3} \pi \):
\[ \frac{4}{3} \pi (r_1^3 + r_2^3) = \frac{1144}{3} \]
Dividing both sides by \( \frac{4}{3} \pi \):
\[ r_1^3 + r_2^3 = \frac{1144}{3} \cdot \frac{3}{4\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r_1^3 + r_2^3 = \frac{1144 \cdot 7}{4 \cdot 22} = \frac{1144 \cdot 7}{88} = 91 \]

### Step 5: Use the Identity for Cubes
We use the identity for the sum of cubes:
\[ r_1^3 + r_2^3 = (r_1 + r_2)(r_1^2 - r_1 r_2 + r_2^2) \]
Substituting \( r_1 + r_2 = 7 \):
\[ 91 = 7(r_1^2 - r_1 r_2 + r_2^2) \]
Dividing both sides by 7:
\[ r_1^2 - r_1 r_2 + r_2^2 = 13 \]

### Step 6: Express \( r_1^2 + r_2^2 \)
Using the square of the sum:
\[ (r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2 \]
Substituting \( r_1 + r_2 = 7 \):
\[ 49 = r_1^2 + r_2^2 + 2r_1 r_2 \]
Now, we can express \( r_1^2 + r_2^2 \):
\[ r_1^2 + r_2^2 = 49 - 2r_1 r_2 \]

### Step 7: Substitute Back
Substituting \( r_1^2 + r_2^2 \) into the earlier equation:
\[ 49 - 2r_1 r_2 - r_1 r_2 = 13 \]
This simplifies to:
\[ 49 - 3r_1 r_2 = 13 \]
So,
\[ 3r_1 r_2 = 49 - 13 = 36 \]
Thus,
\[ r_1 r_2 = 12 \]

### Step 8: Find the Difference of Radii
Using the identity for the difference of squares:
\[ (r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2 \]
Substituting the known values:
\[ (r_1 - r_2)^2 = 49 - 4 \cdot 12 = 49 - 48 = 1 \]
Taking the square root:
\[ r_1 - r_2 = \sqrt{1} = 1 \]

### Final Answer
The difference of the radii \( r_1 \) and \( r_2 \) is:
\[ \boxed{1 \, \text{cm}} \]