Calculus
posted by Marty on .
Find the volume of the cone of maximum value that can be inscribed in a sphere of radius 10cm.

Make a diagram.
let the radius of the cone be x,
let the distance from the base of the cone to the centre of the sphere by y,
then the height of the cone is y+10
draw a line from the centre of the sphere to the end of the radius of the cone.
You now have a rightangled triangle where
x^2 + y^2 = 100
and x^2 = 100  y^2
volume of a cone = (1/3)pi(r^2)h
= (pi/3)(100  y^2)(y+10)
= (pi/3(100y + 1000  y^3  10y^2)
d(volume)/dy = pi/3(100  3y^2  20y)
= 0 for a max/min of volume
so 3y^2 + 20y  100 = 0
(3y10)(y+10) = 0
then y = 10/3 or y = 10 which would be silly
so y = 10/3
sub that back in x^2  100  y^2
and then into the volume of the cone