A spherical shell of lead, whose external diameter is18cm, is melted and recast into a right circular cylinder, whose height is8cm and diameter is12cm. Determine the internal diameter of the shell.

We are given a spherical shell of lead with an external diameter of 18 cm. The shell is melted and recast into a right circular cylinder with a height of 8 cm and a diameter of 12 cm. We need to determine the internal diameter of the shell.

Step 1: Volume of the right circular cylinder
First, we calculate the volume of the cylinder using the formula:

Volume of a cylinder = πr²h

Where:

r = radius of the cylinder
h = height of the cylinder
The diameter of the cylinder is 12 cm, so the radius is:

r = 12 / 2 = 6 cm

The height of the cylinder is given as 8 cm. Thus, the volume of the cylinder is:

Volume of cylinder = π × (6)² × 8 = π × 36 × 8 = 288π cubic centimeters

Step 2: Volume of the spherical shell
The volume of a spherical shell is the difference between the volumes of two spheres: one with the outer radius and one with the inner radius.

The formula for the volume of a sphere is:

Volume of sphere = (4/3)πr³

Let:

R = outer radius of the spherical shell
r = inner radius of the spherical shell
The external diameter of the shell is 18 cm, so the outer radius is:

R = 18 / 2 = 9 cm

The volume of the shell is:

Volume of shell = (4/3)π(R³ - r³)

Substituting the value of R:

Volume of shell = (4/3)π[(9)³ - r³] = (4/3)π[729 - r³]

Step 3: Equating the volumes
Since the lead is melted and recast, the volume of the shell must be equal to the volume of the cylinder:

(4/3)π[729 - r³] = 288π

Canceling π from both sides:

(4/3)[729 - r³] = 288

Multiplying both sides by 3 to eliminate the fraction:

4[729 - r³] = 864

Dividing both sides by 4:

729 - r³ = 216

Now, solving for r³:

r³ = 729 - 216 r³ = 513

Taking the cube root of both sides:

r = ∛513 ≈ 8.04 cm

Step 4: Internal diameter of the shell
The internal diameter of the shell is twice the internal radius:

Internal diameter = 2r = 2 × 8.04 ≈ 16.08 cm

Thus, the internal diameter of the shell is approximately 16.08 cm.