A solid sphere of radius 15 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 13 cm and its height is 180 cm, find the uniform thickness of the cylinder.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1 cm
Given:
- Radius of solid sphere = 15 cm
- External radius of hollow cylinder = 13 cm
- Height of cylinder = 180 cm
Step 1: Volume of the sphere
$V_{\text{sphere}}=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(15)^3 =4500\pi \text{ cm}^3$
Step 2: Volume of the hollow cylinder
Let inner radius = r cm
$V_{\text{cylinder}}=\pi h (R^2 - r^2) =\pi \times 180 \times (13^2 - r^2)$
Since the sphere is melted to form the cylinder:
$180(169 - r^2)=4500$
Divide both sides by 45:
$4(169 - r^2)=100$
$169 - r^2=25$
$r^2=144 \Rightarrow r=12 \text{ cm}$
Step 3: Thickness of the cylinder
$\text{Thickness} = R - r = 13 - 12 = 1 \text{ cm}$