Question

The radii of the bases of two solid right circular cones of same height are r₁ and r₂ respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms r₁, r₂ and R. 

Answer 1

Let the height of the solid cones be 'h'

Volume of solid circular cones 

`V_1 = 1/3pir_1^2h` 

`V_2 = 1/3pir_2^2h` 

Volume of sphere 

= `4/3piR^3` 

Volume of sphere = Volume of cone 1 + volume of cone 2 

`4/3piR^3 = 1/3pir_1^2h + 1/3pir_2^2h` 

`=> 4R^3 = r_1^2h + r_2^2h` 

`=> h(r_1^2 + r_2^2) = (4R)^3` 

`=> h = (4R^3)/((r_1^2 + r_2^2)`