A cylinder is inscribed in a right circular cone of height 2 and radius (at the base) equal to 7.5. What are the dimensions of such a cylinder which has maximum volume?

Question

Radius = 

Height =

Answer 1

If r and h radius and of inscribed cylinder then from similyarity of triangles(2- h)/2 = 2r/15, from here h = 2 - 4r/15.

Volume V = πr²h = πr²(2 - 4r/15)= π(2r² - 4r³/15). Let find derivative and set it to 0:

V' = π(4r - 12r²/15) = 0; we get equation 4r - 4r²/5 = 0, r = 0 and r = 5.

To check if we get maximum we find V'' = π(4 - 8r/5) and at r = 5 V'' = π(4 -8) = -4π < 0 max

Answer: demantions of inscribed cylinder r = 5 and h = 2 - 4·5/15 = 2 - 4/3 = 2/3