Problem 29 - Introductory and Intermediate Algebra for College Students 4th

Question

A cylinder with radius 3 inches and height 4 inches has its radius tripled. How many times greater is the volume of the larger cylinder than the smaller cylinder?

Step-by-Step Solution

Answer

The volume of the larger cylinder is 9 times larger than the volume of the smaller one.

Step 1: Identify given quantities

The smaller cylinder has a radius of \( r_1 = 3 \) inches and a height of \( h = 4 \) inches. The larger cylinder's radius is triple the smaller cylinder's radius, so \( r_2 = 3r_1 = 3*3 = 9 \) inches.

Step 2: Compute the volume of the smaller cylinder

Using the formula for the volume of a cylinder, the volume \( V_1 \) of the smaller cylinder is given by \( V_1 = \pi r_1^2 h = \pi (3^2) 4 = 36\pi \) cubic inches.

Step 3: Compute the volume of the larger cylinder

Using the same formula, the volume \( V_2 \) of the larger cylinder is \( V_2 = \pi r_2^2 h = \pi (9^2) 4 = 324\pi \) cubic inches.

Step 4: Compute the ratio of the two volumes

We are asked how many times greater the volume of the larger cylinder is than the smaller one. We shall compute the ratio \( R \) by the formula \( R = V_2 / V_1 = (324\pi) / (36\pi) = 9 \).