Question

a right circular cylinder and a cone have equal bases and equal Heights if their CSA are in the ratio 8 by 5 show that the ratio between radius to their bases to their height is 3 by 4​

Answer 1

Let the radius and height of cylinder and cone be r and h. 

Let the slant height of cone be l. 

CSA of cylinder : 2πrh

CSA of cone : πrl

Ratio = CSA of cylinder/CSA of cone = 2πrh/πrl = 8/5

⇒ h/l = 4/5 

⇒ h²/l² = 16/25

⇒l² = (25/16)h²

⇒ h² + r² = (25/16)h²

⇒ r² = (9/16)h²

⇒ (r/h)² = (3/4)²

⇒ r/h = 3/4 

hence ratio of radius to height is 3:4

Answer 2

Let the radius and height of cylinder and cone be r and h. 

Let the slant height of cone be l. 

CSA of cylinder : 2πrh

CSA of cone : πrl

CSA of cylinder/CSA of cone = 2πrh/πrl = 8/5

h/l = 4/5 

h²/l² = 16/25

l² = (25/16)h²

h² + r² = (25/16)h²

r² = (9/16)h²

(r/h)² = (3/4)²

r/h = 3/4

Therefore, r : h = 3 : 4