Question

Find the largest volume V of the circular cone that can be inscribed in a sphere of radius = 10 cm

Answer 1

Make a sideview sketch
let the radius of the cone be r cm
let the distance from the base of the cone to the centre of the circle be h
So the height of the cone is h+10 cm

draw a radius from the centre to the base of the cone, it will be a right triangle

r^2 + h^2 = 100
r^2 = 100-h^2

Volume of cone
= (1/3)(pi)r^2 (h+10)
= (1/3)(pi)(100-h^2)(h+10)
= (1/3)(pi)(100h + 1000 -h^3 - 10h^2)
d(volume)/dh
= (1/3)(pi)(100 - 3h^2 - 20h)
= 0 for a max volume

3h^2 + 20h - 100 = 0
(3h - 10)(h + 10) = 0
h = 10/3 cm or a negative
r = √(100 - 100/9) = √800/3
= appr. 9.43 cm

<b>the cone should be 13.33 cm high and have a radius of 9.43 cm
for a volume of
(1/3)(pi)(9.43)^2 (13.33)
= 1241.123 cm^3</b>

I will test by taking r = 9 and r = 9.5, values slightly below and slightly above my answer of r = 9.43
Both volumes should be less than the above
for r = 9, h = 14.359
vol = 1217.96

for r = 9.5, h =
vol = 1240.2

my answer produced a higher volume,
There is a high probability that my answer is correct