Need help with the optimization of volume of 3 dimensional shapes.

A right circular cylinder is inscribed in a cone with height H and base radius R. Find the largest volume of such a cylinder (you should assume they are rotated about the same axis).

If you could please solve in a step by step manner and explain each step, I would greatly appreciate it. Thank you.

Re: Need help with the optimization of volume of 3 dimensional shapes.

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**devilray1018** A right circular cylinder is inscribed in a cone with height H and base radius R. Find the largest volume of such a cylinder (you should assume they are rotated about the same axis).

If you could please solve in a step by step manner and explain each step, I would greatly appreciate it. Thank you.

Have you copied down everything from the question? The largest possible cylinder at the moment is with infinite radius and infinite height...

Re: Need help with the optimization of volume of 3 dimensional shapes.

let the cone be formed by an isosceles triangle with vertex angle on the y-axis and base on the x-axis.

slant of cone in quad I is the line $\displaystyle y = H -\frac{H}{R}x$

radius of the cylinder $\displaystyle r = x < R$

height of the cylinder $\displaystyle h = y = H -\frac{H}{R}x$

$\displaystyle V = \pi r^2 h$

$\displaystyle V = \pi x^2(H -\frac{H}{R}x)$

$\displaystyle V = \pi H (x^2 -\frac{x^3}{R})$

remember H and R are constants, I'll leave you to finish it ...