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
When a solid is melted and recast, its volume does not change. The volume of a hollow cylinder is the volume of the outer cylinder minus that of the hole: \(\pi(R^2-r^2)h\), where \(R\) is the external and \(r\) the internal radius.
The two volumes:
Volume of the sphere \(\displaystyle =\frac{4}{3}\pi\times15^3=\frac{4}{3}\pi\times3375=4500\pi\)
Volume of the hollow cylinder \(=\pi(13^2-r^2)\times180\)
Finding \(r\) (equating the two):
\(180(169-r^2)=4500\)
So \(169-r^2=25\)
\(r^2=144\), so the internal radius \(r=12\) cm
The thickness:
Thickness \(=R-r=13-12=1\) cm
Tip: 12 cm (Option 1) is the internal radius; the question asks for the thickness, the difference between the two radii.