Radius of two cylinders are in the ratio of 2:1 and their volumes are equal. The ratio of their heights will be?

Given:

• The radii of two cylinders are in the ratio 2:1.

• The volumes of the two cylinders are equal.

• We need to find the ratio of their heights.

Concept Used:

The volume of a cylinder is given by the formula:

Volume (V) = π × r² × h

Here, r is the radius and h is the height of the cylinder.

If the volumes are equal, we can equate their formulas and solve for the ratio of their heights.

Calculation:

Let the radii of the two cylinders be r₁ and r₂, and their heights be h₁ and h₂.

From the problem,

r₁ : r₂ = 2 : 1,

so r₁ = 2r₂.

The volumes of the two cylinders are equal, so:

π × r₁² × h₁ = π × r₂² × h₂

Cancel π from both sides:

r₁² × h₁ = r₂² × h₂

Substitute r₁ = 2r₂:

(2r₂)² × h₁ = r₂² × h₂

Simplify:

4r₂² × h₁ = r₂² × h₂

Cancel r₂² from both sides:

4h₁ = h₂

Rearrange to get the ratio of heights:

h₁ : h₂ = 1 : 4

Conclusion:

∴ The ratio of their heights is 1:4.