A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 3.5. What are the dimensions of such a cylinder which has maximum volume?

Question

Please give the radius and the height.

Answer 1

Let x is radius of cylinder and h is height. Then volume v = πx²h. Now (3.5 - x)/3.5 = h/5.5

5.5(3.5 - x) = 3.5h; ; h = 11(3.5 - x)/7; v = 11π/7(3.5x² - x³); v' = 11π/7(7x - 3x²) = 0; x = 0 or x = 7/3;

v'' = 11π/7(7 - 6x); v''(7/3) = 11π/7(7 - 14) = - 11π < 0. So, we have max

Answer: x = 7/3 and h = 11/7(3.5 - 7/3) = 11/7·3.5/3 = 11/6