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Find radius of a cylinder with the biggest area surface inscribed in cone. Cone has radius R, and height H. (sorry for bad english)
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i've tried and got this:
$x = (hr)/2(h-r)$, but what if h = r?

So, I want you to imagine a cylinder inside a cone of radius $r$ and height $h$ and a cylinder having a radius say $x$. Now imagine that the cylinder has a certain height and I am going to ignore that, instead, I am going to take the remaining height and call it $h^{'}$ so we get the height of the cylinder as $h-h^{'}$. Now leave that aside for a minute. Now we will use the same relation you derived, $$frac{h^{'}}{x} = frac{h}{r}$$ So, $$h^{'} = frac{hx}{r}$$ Now, the surface area $S_a$ is, $$ S_ a = 2 pi x Bigl(h-h^{'}Bigl) + 2pi x^2$$ which is equivalent to, $$ S_ a = 2 pi x Bigl(h-frac{hx}{r}Bigl) + 2pi x^2 $$ which gives us, $$S_a = 2 pi h Bigl(x - frac{x^2}{r}Bigl) + 2pi x^2 $$ Now in order to find the maximum area that can be described in the cone, we have to differentiate w.r.t $x$ and then set to $0$.

Now, $$frac{dS_{a}}{dx}=0$$ for maximum area. $$2 pi h cdot frac{dBigl(x - frac{x^2 }{r}Bigl)}{dx} + 2pi frac{dx^2}{dx}=0$$ Which gives us, $$2 pi h Bigl(1 - frac{2x}{r}Bigl) + 4pi x=0$$ So, $$hBigl(1- frac{2x}{r} Bigl) + 2x = 0$$ and now using this relation you can find $x$ in terms of $h$ and $r$. Personally I would only use, the curved surface area becuase the solution is much nicer in that case, but whatever, to each his own.