By melting a solid lead sphere of diameter 12 cm, three small sphere are made whose diameters are in the ratio 3:4:5. The radius (in cm) of the smallest sphere is

Options

A. 3
B. 6
C. `1.5`
D. 4

Answer

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

### Step 1: Find the radius of the large sphere.
The diameter of the large sphere is given as 12 cm. The radius (R) can be calculated using the formula:
\[ R = \frac{\text{Diameter}}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm} \]

### Step 2: Calculate the volume of the large sphere.
The volume (V) of a sphere is calculated using the formula:
\[ V = \frac{4}{3} \pi R^3 \]
Substituting the radius we found:
\[ V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

### Step 3: Set up the ratios for the diameters of the smaller spheres.
The diameters of the three smaller spheres are in the ratio 3:4:5. Let the diameters be:
- \(d_1 = 3x\)
- \(d_2 = 4x\)
- \(d_3 = 5x\)

### Step 4: Calculate the radii of the smaller spheres.
The radii (r) of the smaller spheres can be calculated as:
- \(r_1 = \frac{d_1}{2} = \frac{3x}{2}\)
- \(r_2 = \frac{d_2}{2} = \frac{4x}{2} = 2x\)
- \(r_3 = \frac{d_3}{2} = \frac{5x}{2}\)

### Step 5: Write the equation for the volumes of the smaller spheres.
The total volume of the three smaller spheres must equal the volume of the large sphere:
\[ \frac{4}{3} \pi r_1^3 + \frac{4}{3} \pi r_2^3 + \frac{4}{3} \pi r_3^3 = 288 \pi \]
Substituting the expressions for the radii:
\[ \frac{4}{3} \pi \left(\frac{3x}{2}\right)^3 + \frac{4}{3} \pi (2x)^3 + \frac{4}{3} \pi \left(\frac{5x}{2}\right)^3 = 288 \pi \]

### Step 6: Simplify the equation.
Cancel \( \frac{4}{3} \pi \) from both sides:
\[ \left(\frac{3x}{2}\right)^3 + (2x)^3 + \left(\frac{5x}{2}\right)^3 = 216 \]
Calculating each term:
\[ \frac{27x^3}{8} + 8x^3 + \frac{125x^3}{8} = 216 \]
Combine the terms:
\[ \left(\frac{27x^3 + 64x^3 + 125x^3}{8}\right) = 216 \]
\[ \frac{216x^3}{8} = 216 \]
Multiply both sides by 8:
\[ 216x^3 = 1728 \]
Divide by 216:
\[ x^3 = 8 \]
Taking the cube root:
\[ x = 2 \]

### Step 7: Find the radius of the smallest sphere.
The radius of the smallest sphere is:
\[ r_1 = \frac{3x}{2} = \frac{3 \cdot 2}{2} = 3 \text{ cm} \]

### Final Answer:
The radius of the smallest sphere is **3 cm**.

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