Optimization: A cylinder is inscribed in a right

Question

A cylinder is inscribed in a right circular cone of height 3.5 and radius (at the base) equal to 4.5. What are the dimensions of such a cylinder which has maximum volume

Answer 1

The numbers in this problem get messy.

Let h be the height of the cylinder and r its radius.

Then h/3.5 = (2.25-r)/2.25 and

V=πr²h

Substitution of the expression for h into the expression for V yields a cubic in r.

Differentiation of V with respect to r eventually gives 3r²=9/4.

You should do the substitution and follow the algebra to the solution from here.