Question

The ratio of the radius and the height of a solid metallic right circular cylinder is 7 : 27. This is melted and made into a cone of diameter 14 cm and slant height 25 cm. Find the height of the:

1. cone
2. cylinder

Answer 1

Given: The ratio of radius to height of the cylinder = 7 : 27.

The cone formed has diameter 14 cm, so radius R = 7 cm and slant height l = 25 cm.

Step-wise calculation:

1. Find the height of the cone using l² = R² + h_cone²: 

`h_"cone" = sqrt(l^2 - R^2)`

= `sqrt(25^2 - 7^2)`

= `sqrt(625 - 49)`

= `sqrt(576)`

= 24 cm

2. Let the common factor for the cylinder be k.

Then cylinder radius = 7k and cylinder height = 27k.

Volume of cylinder = π(7k)²(27k)

= π × 49 × 27 × k³

= 1323 πk³

3. Volume of the cone = `1/3 πR^2h_"cone"`

= `1/3 π xx 7^2 xx 24`

= `1/3 π xx 49 xx 24`

= 392π

4. Equate volumes metal conserved:

1323πk³ = 392π 

⇒ 1323k³ = 392

Note 1323 = 27 × 49 and 392 = 8 × 49 

So, `k^3 = (8 xx 49)/(27 xx 49)` 

= `8/27`

⇒ `k = 2/3`

5. Cylinder height = 27k

= `27 xx 2/3` 

= 18 cm