A cylinder is inscribed in a right circular cone of height 7.5 and radius (at the base) equal to 7. What are the dimensions of such a cylinder which has maximum volume?

Answer 1

The first thing that we have to notice is that we can express the radius of the cylinder as a function of its height because it is contained by the dimensions of the cone.

Look at a 2-D cross section of the cone with the cylinder inside and you have a right triangle and a rectangle contained within its boundaries. The length of the sides of the triangle are 7.5 and 7 while the length of the sides of the rectangle are r and h. Because of the rules for similar triangles we can show that:

7.5/7 = h/(7-r)

h = 7.5(7-r)/7

We can now express the volume of the cylinder in terms of r only

V(r) = π r² (7.5 - (7.5/7) r)

Take the derivative with respect to r and set equal to zero.

dV/dr = 15π r -3(7.5/7)π r² = 0

r = 0, 14/3

r = 0 is the minimum volume

r = 14/3 is the maximum volume

V(14/3) = (490/9) π with a radius of 14/3 and a height of 5/2