A cylinder and a cone have equal radii of their bases and equal height

1.	A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8:5, show that the radius and height of each has the ratio 3:4.
Answer» Consider the curved surface area of cylinder and cone as 8x and 5x. So we get 2 πrh = 8x …….. (1) Πr √ (h² + r²) = 5x ……… (2) By squaring equation (1) (2 πrh)² = (8x)² So we get 4 π²r²h² = 64 x² …….. (3) By squaring equation (2) Π²r² (h² + r²) = 25x² …… (4) Dividing equation (3) by (4) 4 π²r²h²/ π²r² (h² + r²) = 64 x²/25x² On further calculation h²/ (h² + r²) = 16/25 It can be written as 9 h² = 16 r² So we get r²/ h² = 9/16 By taking square root r/ h = ¾ We get r: h = 3:4 Therefore, it is proved that the radius and height of each has the ratio 3:4.

A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8:5, show that the radius and height of each has the ratio 3:4.

Answer»

Consider the curved surface area of cylinder and cone as 8x and 5x.

So we get

2 πrh = 8x …….. (1)

Πr √ (h² + r²) = 5x ……… (2)

By squaring equation (1)

(2 πrh)² = (8x)²

So we get

4 π²r²h² = 64 x² …….. (3)

By squaring equation (2)

Π²r² (h² + r²) = 25x² …… (4)

Dividing equation (3) by (4)

4 π²r²h²/ π²r² (h² + r²) = 64 x²/25x²

On further calculation

h²/ (h² + r²) = 16/25

It can be written as

9 h² = 16 r²

So we get

r²/ h² = 9/16

By taking square root

r/ h = ¾

We get

r: h = 3:4

Therefore, it is proved that the radius and height of each has the ratio 3:4.