The surface area of a sphere of radius `5 cm` is five times the curved surface area of a cone of radius `4 cm.` Find the height and volume (correct to two decimal places) of the cone.

Explanation

Surface area of the given sphere
`4pi R^(2) = (4pi xx 5 xx 5) cm^(2) = (100pi) cm^(2)`
Radius of the base of the cone, `r = 4 cm`
Let the slant height of the cone be `lcm`. Then,
curved surface area of the given cone
`= (pi rl) = (piu xx 4 xx l) cm^(2) = (4pil) cm^(2)`
`:. 100pi = 5 xx (4pil) rArr l = 5 cm`
Let the height of the cone `h cm.` Then,
`l^(2) = h^(2) + r^(2) rArr h^(2) = (l^(2) - r^(2)) = (5^(2) - 4^(2)) = 9 rArr h = 3cm`
Volume of the cone `= (1)/(3) pi r^(2)h`
`=((1)/(3) xx 3.14 xx 4 xx 4 xx 3) cm^(3) = 50.24 cm^(3)`
Hence, the height of the cone is 3 cm and its volume is `50.24 cm^(3)`