The radii of the two cones are in the ratio of 2 : 5 and their volumes are in…

To find the ratio of the heights of the two cones, we can use the formula for the volume of a cone, which is V = (1/3)πr²h.

Step-by-Step Analysis
Define the Variables:

Let the radii of the two cones be r₁ and r₂. We are given r₁/r₂ = 2/5.

Let the heights be h₁ and h₂. We want to find the ratio h₁/h₂.

Let the volumes be V₁ and V₂. We are given V₁/V₂ = 3/5.

Set up the Equation:
Using the volume formula:
V₁/V₂ = [(1/3)π(r₁)²h₁] / [(1/3)π(r₂)²h₂]
Since (1/3)π is common to both, it cancels out:
V₁/V₂ = (r₁/r₂)² * (h₁/h₂)

Substitute and Solve:
Substitute the known ratios into the equation:
3/5 = (2/5)² * (h₁/h₂)
3/5 = (4/25) * (h₁/h₂)

To isolate (h₁/h₂), multiply both sides by (25/4):
(h₁/h₂) = (3/5) * (25/4)
(h₁/h₂) = (3 * 25) / (5 * 4)
(h₁/h₂) = 75 / 20
Simplify by dividing by 5:
(h₁/h₂) = 15 / 4

The ratio of their heights is 15:4.