Area/volume geometry word problem

Question

A right square prism four feet tall drops snugly through a ten inch square hole. The contents of the prism must be transferred to a right circular cylinder that must barley fit through the same hole. How tall must the cylinder be?

Answer 1

Hello Kim,

The volume of the right-square prism may be expressed as

V = Ah,

where A is the cross-sectional area and h is the height. Since the cross-section is square, the area is

A = x², where x is the side-length of the square. Hence, we may write the volume as

V = x²h.

The prism fits snugly through a square hole 10 inches on a side, so this implies x = 10 inches. We are also given that h = 4ft = 48inches, so

V = 10²*48 = 4800 in³.

(Volume given in cubic inches.)

The volume of a right-circular cylinder with radius r and height H is

V = πr²H.

If the right-circular cylinder fits through the same hole as the prism, then r = 5. So we set the volume of the cylinder equal to the volume of the prism to find the height of the cylinder.

πr²H = 4800

π5²H = 4800

25πH = 4800.

Therefore,

H = 4800/(25π) = 192/π, or

H ≅ 61.1 inches.

Hope that helps! Let me know if you need any clarification.

William