Problem:
The volume of two hemispheres are in ratio 27: 125. Find the ratio of their radii.

Solution:

Step 1: Understanding the problem
We are given that the volume of two hemispheres is in the ratio 27: 125. We need to find the ratio of their radii.

Step 2: Understanding the formula for the volume of a hemisphere
The volume of a hemisphere is given by the formula:
V = (2/3) * π * r³
where V is the volume and r is the radius of the hemisphere.

Step 3: Setting up the equation
Let's assume the radii of the two hemispheres are r₁ and r₂ respectively.
Given that the volumes of the hemispheres are in the ratio 27: 125, we can write:
(2/3) * π * r₁³ / (2/3) * π * r₂³ = 27 / 125.

Step 4: Simplifying the equation
Simplifying the equation, we get:
r₁³ / r₂³ = 27 / 125.

Step 5: Taking the cube root of both sides
Taking the cube root of both sides of the equation, we get:
(r₁ / r₂)³ = (27 / 125).

Step 6: Solving for the ratio of radii
Taking the cube root of both sides, we find:
(r₁ / r₂) = (3 / 5).

Therefore, the ratio of the radii of the two hemispheres is 3: 5.

Conclusion:
The ratio of the radii of the two hemispheres is 3: 5.