A right circular cylinder is inscribed in a cone with height h and base radius r. Find the
largest possible surface area of such a cylinder.

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  1. the the radius of the cylinder by x and the height of the cylinder be y.
    (the h and r will be constants)

    by ratios : x/(h-y) = r/h
    y = (hr - hx)/r

    I will assume that you want both top and bottom of the cylinder included in your surface area, if not you will have to change the equation.

    SA = 2πx^2 +2πxy
    = 2πx^2 + 2πhx - (2πh/r)x^2 after subbing in the above y

    SA' = 4πx + 2πh - (4πh/r)x
    = 0 for a max/min of SA

    I get x = hr/(2h-2r)

    put that back into SA = ....

    I will let you finish the algebra.

    (Also check my steps, I tend to make typing errors)

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