Find the largest total surface area (including top and bottom) of a cylinder inscribed in a cone of base radius 9 and height 21.

We need to maximize the following function https://webwork.math.ohio-state.edu/...79cb7891c1.png and https://webwork.math.ohio-state.edu/...11a2eaba01.png:

https://webwork.math.ohio-state.edu/...c37eeb1e11.png

Then we solve for https://webwork.math.ohio-state.edu/...79cb7891c1.png in terms of https://webwork.math.ohio-state.edu/...11a2eaba01.png obtaining:

https://webwork.math.ohio-state.edu/...72a570cfa1.png

We thus obtain the following formula for https://webwork.math.ohio-state.edu/...f3fe445111.png in terms of https://webwork.math.ohio-state.edu/...11a2eaba01.png alone:

https://webwork.math.ohio-state.edu/...c37eeb1e11.png

https://webwork.math.ohio-state.edu/...11a2eaba01.png varies over the interval https://webwork.math.ohio-state.edu/...4d9569d581.png, where

https://webwork.math.ohio-state.edu/...ea9131f391.png

https://webwork.math.ohio-state.edu/...7973f0e8c1.png

https://webwork.math.ohio-state.edu/...f3fe445111.png has one stationary point at:

https://webwork.math.ohio-state.edu/...b10677e021.png

We conclude that the maximum possible total surface area of such a cylinder is:

https://webwork.math.ohio-state.edu/...7a3e4ac791.png

I know that the surface area is S=2pir(r+h)