The sum and difference of the slant height and height of cone A are in the ratio 3 ∶ 1, while the sum and difference of the slant height and height of cone Y are in the ratio 5 ∶ 1. If the volume of cone A is 30% of the volume of cone B, then the height of cone A is what percentage of the height of cone B.

The sum and difference of the slant height and height of cone A are in the ratio 3 ∶ 1, while the sum and difference of the slant height and height of cone Y are in the ratio 5 ∶ 1. If the volume of cone A is 30% of the volume of cone B, then the height of cone A is what percentage of the height of cone B. Correct Answer 50%

Given:

Sum and difference of the slant height and height of cone A are in the ratio 3 ∶ 1

Sum and difference of the slant height and height of cone Y are in the ratio 5 ∶ 1

Volume of cone A is 30% of the volume of cone B

Formula used:

Volume of cone = (1/3) πr²h

Calculation:

Let the radius, slant height and height of cone A be r, l and h units respectively

Given, (l + h)/(l – h) = 3

⇒ l + h = 3l – 3h

⇒ 2l = 4h

⇒ l = 2h

For cone A, l² = h² + r²

⇒ (2h)² = h² + r²

⇒ 3h² = r²

⇒ Volume of cone A = (1/3) πr²h = (1/3) π × 3h² × h = πh³ cubic units

Similarly,

Let the radius, slant height and height of cone B be ‘R’, ‘L’ and ‘H’ units respectively

Now, (L + H)/(L – H) = 5

⇒ L + H = 5L – 5H

⇒ 4L = 6H

⇒ L = 3H/2

For cone B, L² = H² + R²

⇒ (3H/2)² = H² + R²

⇒ 9H² = 4H² + 4R²

⇒ 5H²/4 = R²

⇒ Volume of cone B = (1/3) πR²H = (1/3) π × 5H²/4 × H = (5/12) πH³ cubic units

Now,

Volume of cone A = 30% of volume of cone B

⇒ πh³ = (3/10) × (5/12) πH³

⇒ h³/H³ = 1/8

⇒ h/H = 1/2