A cylinder is inscribed in a right circular cone of height 6.5 and radius (at the base) equal to 8. What are the dimensions of such a cylinder which has maximum volume?
Radius =
Height =
Suppose r is the base radius of the cylinder inscribing the cone. Let h be the height.
Using similar triangles on the largest cross-section,
h8−r=6.58h=132−1316r
Then the volume is πr2h=1316πr2(8−r).
Solving ddr1316πr2(8−r)=0 gives r = 16/3 or r = 0 (rej.)
Note that
d2dr2|r=1631316πr2(8−r)=−13π<0. Then maximum volume is attained when r = 16/3.
Now, you can calculate the height and get your answer.