The radii of the internal and external surfaces of a metallic spherical shell are 3 cm and 5 cm respectively. It is melted and recast into a solid right circular cone of height 32 cm. Find the diameter of the base of the cone.

Answer

To solve the problem, we need to find the diameter of the base of a cone formed from a metallic spherical shell with given internal and external radii. Here’s the step-by-step solution:

### Step 1: Calculate the Volume of the Spherical Shell
The volume of a spherical shell is given by the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
where \( R \) is the external radius and \( r \) is the internal radius.

Given:
- External radius \( R = 5 \, \text{cm} \)
- Internal radius \( r = 3 \, \text{cm} \)

Substituting the values:
\[ V = \frac{4}{3} \pi (5^3 - 3^3) \]
Calculating \( 5^3 \) and \( 3^3 \):
\[ 5^3 = 125 \quad \text{and} \quad 3^3 = 27 \]
Now substituting these values:
\[ V = \frac{4}{3} \pi (125 - 27) = \frac{4}{3} \pi (98) \]

### Step 2: Calculate the Volume of the Cone
The volume of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r_c^2 h \]
where \( r_c \) is the radius of the base of the cone and \( h \) is the height of the cone.

Given:
- Height \( h = 32 \, \text{cm} \)

Setting the volume of the cone equal to the volume of the spherical shell:
\[ \frac{4}{3} \pi (98) = \frac{1}{3} \pi r_c^2 (32) \]

### Step 3: Simplify the Equation
We can cancel \( \pi \) and \( \frac{1}{3} \) from both sides:
\[ 4 \times 98 = r_c^2 \times 32 \]
Calculating \( 4 \times 98 \):
\[ 392 = r_c^2 \times 32 \]

### Step 4: Solve for \( r_c^2 \)
Now, divide both sides by 32:
\[ r_c^2 = \frac{392}{32} \]
Simplifying \( \frac{392}{32} \):
\[ r_c^2 = \frac{49}{4} \]

### Step 5: Find \( r_c \)
Taking the square root of both sides:
\[ r_c = \sqrt{\frac{49}{4}} = \frac{7}{2} \, \text{cm} \]

### Step 6: Calculate the Diameter of the Cone
The diameter \( d_c \) of the cone is given by:
\[ d_c = 2r_c = 2 \times \frac{7}{2} = 7 \, \text{cm} \]

### Final Answer
The diameter of the base of the cone is \( 7 \, \text{cm} \).

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