Using $$100\pi cm^{3}$$ volume of clay, a hollow cylinder of height 20 cm and thickness 2 cm, has been made. What is the capacity of the cylinder?
Solution
We can take the hollow cylinder to be made of two concentric cylinders, with the inner cylinder carved out of the outer cylinder.
Taking the radius of the outer cylinder to be R, we know that the thickness of the cylinder is 2 cm; therefore, the radius of the inner cylinder will be R-2 cm.
We know that the volume of this hollow cylinder would be $$100\pi\ $$
$$\pi\ R^2h-\pi\ \left(R-2\right)^2h=100\pi\ $$
Putting h=20 and solving the left-hand side, we get
$$4R\ -4=5$$
$$R=\frac{9}{4}$$
The inner radius would be $$\frac{9}{4}-2=\frac{1}{4}$$ cm
The capacity of this hollow cylinder would be $$\pi\ \left(\frac{1}{4}\right)^220=\pi\ \frac{20}{16}=\frac{5\pi}{4}\ cm\ $$
Therefore, Option C is the correct answer.
