A metallic sphere of radius 12 cm is melted into three smaller spheres. If the radii of two smaller spheres are 6 cm and 8 cm, the radius of the third is

Options

A. 14 cm
B. 16 cm
C. 10 cm
D. 12 cm

Answer

To solve the problem, we need to find the radius of the third sphere after melting a larger sphere into three smaller spheres. Here’s a step-by-step solution:

### Step 1: Calculate the volume of the larger sphere
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the larger sphere with a radius of 12 cm:
\[ V_{large} = \frac{4}{3} \pi (12)^3 \]
Calculating \(12^3\):
\[ 12^3 = 1728 \]
Now substituting back into the volume formula:
\[ V_{large} = \frac{4}{3} \pi (1728) = \frac{6912}{3} \pi = 2304 \pi \text{ cm}^3 \]

### Step 2: Calculate the volumes of the two smaller spheres
For the first smaller sphere with a radius of 6 cm:
\[ V_{small1} = \frac{4}{3} \pi (6)^3 \]
Calculating \(6^3\):
\[ 6^3 = 216 \]
Now substituting back into the volume formula:
\[ V_{small1} = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \text{ cm}^3 \]

For the second smaller sphere with a radius of 8 cm:
\[ V_{small2} = \frac{4}{3} \pi (8)^3 \]
Calculating \(8^3\):
\[ 8^3 = 512 \]
Now substituting back into the volume formula:
\[ V_{small2} = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi = 682.67 \pi \text{ cm}^3 \]

### Step 3: Set up the equation for the volume of the third sphere
The total volume of the three smaller spheres must equal the volume of the larger sphere:
\[ V_{large} = V_{small1} + V_{small2} + V_{small3} \]
Substituting the known volumes:
\[ 2304 \pi = 288 \pi + 682.67 \pi + V_{small3} \]

### Step 4: Solve for the volume of the third sphere
Combining the volumes of the first two smaller spheres:
\[ 2304 \pi = (288 + 682.67) \pi + V_{small3} \]
Calculating the sum:
\[ 288 + 682.67 = 970.67 \]
Now substituting back:
\[ 2304 \pi = 970.67 \pi + V_{small3} \]
Subtracting \(970.67 \pi\) from both sides:
\[ V_{small3} = 2304 \pi - 970.67 \pi = 1333.33 \pi \text{ cm}^3 \]

### Step 5: Calculate the radius of the third sphere
Using the volume formula for the third sphere:
\[ V_{small3} = \frac{4}{3} \pi r_3^3 \]
Setting this equal to the volume we found:
\[ 1333.33 \pi = \frac{4}{3} \pi r_3^3 \]
Dividing both sides by \(\pi\):
\[ 1333.33 = \frac{4}{3} r_3^3 \]
Multiplying both sides by \(\frac{3}{4}\):
\[ r_3^3 = 1333.33 \times \frac{3}{4} = 1000 \]
Taking the cube root:
\[ r_3 = \sqrt[3]{1000} = 10 \text{ cm} \]

### Final Answer
The radius of the third sphere is **10 cm**.

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