The height of one cone is 3 times the height of another cone, while its radius is half of the radius of other cone. If their total volume is 100 unit³ , then the difference in the volumes of the cones is ______ unit³ .

Given:

Let the height of the first cone be H₁ and the height of the second cone be H₂.

Let the radius of the first cone be R₁ and the radius of the second cone be R₂.

H₁ = 3 × H₂, R₁ = (1/2) × R₂.

Total volume of the two cones = 100 unit³

Formula used:

Volume of a cone = (1/3)πR²H

Let V₁ and V₂ be the volumes of the first and second cone respectively.

Calculation:

Volume of the first cone:

⇒ V₁ = (1/3)π(R₁)²(H₁)

⇒ V₁ = (1/3)π((1/2) × R₂)²(3 × H₂)

⇒ V₁ = (1/3)π(1/4)R₂²(3H₂)

⇒ V₁ = (1/4) × 3 × (1/3)πR₂²H₂

⇒ V₁ = (3/4) × V₂

Total volume:

⇒ V₁ + V₂ = 100

⇒ (3/4)V₂ + V₂ = 100

⇒ (7/4)V₂ = 100

⇒ V₂ = 100 × (4/7)

⇒ V₂ = 400/7 ≈ 57.14 unit³

Volume of the first cone:

⇒ V₁ = (3/4) × V₂ = (3/4) × 400/7 ≈ 42.86 unit³

Difference in the volumes of the cones:

⇒ V₂ - V₁ = 57.14 - 42.86 ≈ 14.28 unit³

∴ The difference in the volumes of the cones is approximately 14.3 unit³.