The ratio of the heights of two right circular cones is 25:64 and the ratio of…

For a right circular cone, volume V = (1/3)πr2h, where r is the base radius and h is the height; so for two cones, the ratio of their volumes equals the ratio of (radius2 × height), because the constant factor (1/3)π is the same for both cones and cancels out. Since a radius is always half its diameter, two cones' radii are in the same ratio as their diameters.

1. The diameter ratio is 4:5, so the radius ratio r1:r2 is also 4:5, since radius equals diameter divided by 2, which does not change the ratio.

2. The height ratio is given directly as h1:h2 = 25:64.

3. The volume ratio V1:V2 = r12h1 : r22h2 = (42 × 25) : (52 × 64) = (16 × 25) : (25 × 64).

4. The common factor 25 cancels from both sides, leaving 16:64, which reduces, dividing both parts by 16, to 1:4.

As a check, take r1 = 4, r2 = 5, h1 = 25, and h2 = 64, consistent with both given ratios. Then V1 = (1/3)π(42)(25) = 400π/3 and V2 = (1/3)π(52)(64) = 1600π/3, so V1:V2 = 400:1600 = 1:4, the same result, confirming the volume ratio is 1:4.