Mark the correct alternative of the following.
The height of sand in a cylindrical-shaped can drop by $$3$$ inches when $$1$$ cubic foot of sand is pound out. What is the diameter of the cylinder (in inches)?
A
$$\dfrac{24}{\sqrt{\pi}}$$
B
$$\dfrac{32}{\sqrt{\pi}}$$
C
$$\dfrac{48}{\sqrt{\pi}}$$
Correct option is B. $$\dfrac{48}{\sqrt{\pi}}$$
Let $$r$$ be the radius of the cylinder.
It is given that, the height drops $$3$$ inches, when $$1$$ cubic foot of sand is poured out and $$1$$ foot $$=12\ \text{in}$$
So,
Volume of reduced cone = Volume of 1 cubic foot sand
$$\pi r^2 h=12\times 12\times 12\times 12$$
$$\pi r^2 \times 3=12\times 12\times 12$$
$$r^2=\dfrac{12\times 12\times 4}{\pi}$$
$$r=\dfrac{12\times 2}{\sqrt{\pi}}$$
$$\therefore$$ $$r=\dfrac{24}{\sqrt{\pi}}$$
The diameter of cylinder $$=2r=\dfrac{48}{\sqrt{\pi}}$$