Question

Show that the right circular cone of least curved surface and given volume has an altitude equal to(2)^1/2 time the radius of the base.

Answer 1

HELLO DEAR,

let r be the radius of the base.
h be the height and l be the slant height of cone.

now,

V = πr²h/3
h = 3V/πr²-------( 1 )

surface area (s) = πrl

on squaring both side,

s² = π²r²l²

s² = π²r²(h² + r²) [as , l² = h² + r²]

s² = π²r²(9V²/π²r⁴ + r²)

s² = 9V²/πr² + π²r⁴

now,

2s *ds/dr = -18V²/r³ + 4π²r³

now, ds/dr = 0

4π²r³ = 18V²/r³

2π²r³ = 9V²/r³

2π²r^6 = 9(1/9 * π²r⁴h²) -----from( 1 )

2r² = h²

h = 2√r

hence, the right circular cone of least curved surface and given volume has an altitude equal to(2)^1/2 time the radius of the base.

I HOPE ITS HELP YOU DEAR,
THANKS