# Find radius of a cylinder with the biggest area surface inscribed in cone

Find radius of a cylinder with the biggest area surface inscribed in cone. Cone has radius R, and height H. (sorry for bad english) enter image description here

i've tried and got this: $$x = (hr)/2(h-r)$$, but what if h = r?

• What have you tried? Nov 10 '18 at 16:20
• Hello @3a43mka, welcome to MSE. Can you please add your own attempts in the question. Nov 10 '18 at 16:20
• Added my attempts> Nov 10 '18 at 16:42
• I am assuming you are talking about only the curved surface area and not including the top and bottom of the cylinder? Nov 10 '18 at 17:13
• Nope, full surface area Nov 10 '18 at 21:17

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, $$S_ a = 2 \pi x \Bigl(h-h^{'}\Bigl) + 2\pi x^2$$ which is equivalent to, $$S_ a = 2 \pi x \Bigl(h-\frac{hx}{r}\Bigl) + 2\pi x^2$$ which gives us, $$S_a = 2 \pi h \Bigl(x - \frac{x^2}{r}\Bigl) + 2\pi x^2$$ 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. $$2 \pi h \cdot \frac{d\Bigl(x - \frac{x^2 }{r}\Bigl)}{dx} + 2\pi \frac{dx^2}{dx}=0$$ Which gives us, $$2 \pi h \Bigl(1 - \frac{2x}{r}\Bigl) + 4\pi x=0$$ So, $$h\Bigl(1- \frac{2x}{r} \Bigl) + 2x = 0$$ 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.