Question

The total surface area of a right circular cone of slant height 13 cm is 90π cm².

Calculate:  

1. its radius in cm.
2. its volume in cm³. [Take π = 3.14].

Answer 1

Total surface area of cone = 90π cm²

Slant height (l) = 13 cm

i. Let r be its radius, then 

Total surface area = πrl + πr² = πr(l + r) 

∴ πr(l + r) = 90π 

`=>` r(13 + r) = 90 

`=>` r² + 13r – 90 = 0

`=>` r² + 18r – 5r – 90 = 0

`=>` r(r + 18) – 5(r + 18) = 0 

`=>` (r + 18)(r – 5) = 0 

Either r + 18 = 0, then r = –18 which is not possible 

or r – 5 = 0, then r = 5  

Therefore, radius = 5 cm 

ii. Now 

`h = sqrt(l^2 - r^2)`

= `sqrt (13^2 - 5^2)`

= `sqrt (169 - 25)`

= `sqrt(144)`

h = 12 cm

Volume = `1/3pir^2h`

= `1/3 xx 3.14 xx 5 xx 5 xx 12`

= 314 cm³