Question

The curved surface area of a right circular cone is half of another right circular cone. If the ratio of their slant heights is 2 : 1 and that of their volumes is 3 : 1, find ratio of their:

1. radii
2. heights

Answer 1

Given:

Let cone 1 be the cone whose curved surface area is half of cone 2.

Curved surface area: π r l. Volume: `1/3 πr^2h`.

`(CSA_1)/(CSA_2) = 1/2` 

l₁ : l₂ = 2 : 1

V₁ : V₂ = 3 : 1

Step-wise calculation:

1. From curved surface areas: `(r_1l_1)/(r_2l_2) = 1/2`. 

With `l_1/l_2 = 2` 

`r_1/r_2 xx 2 = 1/2`

⇒ `r_1/r_2 = (1/2)/2 = 1/4`

2. From volumes: `(r_1^2h_1)/(r_2^2h_2) = 3`. 

Substitute `(r_1/r_2)^2 = (1/4)^2`

= `1/16 : (1/16) xx (h_1/h_2)`

= 3 

⇒ `h_1/h_2 = 3 xx 16`

= 48

3. The slant relation l² = r² + h² imposes a value for `R^2/H^2` but does not change the ratios found above; the three given conditions are compatible.

1. Ratio of radii r₁ : r₂ = 1 : 4.
2. Ratio of heights h₁ : h₂ = 48 : 1.