Two cones have their volumes in the ratio 1 to 10, while their radii are in the ratio 1 to 2. What is the ratio of their heights?

Explanation

The volume of a cone is given by (1/3)πr²h. Since π/3 is constant, the volume ratio depends on r²h. Given volume ratio V1/V2 = 1/10 and radius ratio r1/r2 = 1/2, substituting gives (1)²h1 / (2)²h2 = 1/10, which simplifies to h1/h2 = 2/5. Therefore, the height ratio is 2 : 5.