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Math Help - Need help with the optimization of volume of 3 dimensional shapes.

  1. #1
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    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.
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    Re: Need help with the optimization of volume of 3 dimensional shapes.

    Quote Originally Posted by devilray1018 View Post
    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...
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    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 y = H -\frac{H}{R}x

    radius of the cylinder r = x < R

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

    V = \pi r^2 h

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

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

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