Final answer:
The ratio of the heights of the two cylinders, given the volume ratio of 27:16 and the radius ratio of 3:2, is found to be 3:4 using the volume formula of a cylinder.
Step-by-step explanation:
The ratio of the volumes of two cylinders is given as 27:16, and the ratio of their radii is 3:2. To find the ratio of their heights, we use the formula for the volume of a cylinder, V = πr²h, where V represents the volume, r the radius, and h the height. Since the volumes are proportional to the product of the square of the radius and the height of the cylinders, we can express the ratio of the volumes (V1/V2) as (r1²h1) / (r2²h2).
Given the ratios of the radii (r1:r2) and volumes (V1:V2), we have (3/2)² = 9/4 and 27/16, respectively. Setting up the equation (9/4)(h1/h2) = 27/16, we need to solve for the ratio h1/h2. Upon simplifying the equation, we find that h1/h2 = (27/16) / (9/4) = (27/16) ∙ (4/9) = 3/4, so the ratio of their heights is 3:4.