right circular cylinder inscribed in a sphere

Question

Find the dimensions of the right-circular cylinder of greatest vloume that can be inscribed in a sphere with a radius of 6 $in$

I think I need help visualizing, and maybe the solution.

I've already read this, but somehow it is not much of a help Here

Thank you!

Answer 1

I think I need help visualizing, ..

(Large version)

and maybe the solution..

One approach is to come up with a model of the inscribed cylinder, which allows to determine its volume $V$ for a given height $h$, the cylinder ranging from $-h$ to $h$ in $z$-direction, and then maximize $V(h)$.

The volume of the cylinder is \begin{align} V &= A H \\ &= \pi R^2 H \end{align} where the rim of the cylinder is part of the sphere and fulfills $$ x^2 + y^2 + z^2 = 6^2 \Rightarrow \\ \underbrace{x^2 + y^2}_{R^2} = 6^2 - z^2 $$ Note that $H = 2h$, as $h$ is the $z$-coordinate of the top surface of the cylinder. We get $$ V(h) = \pi (36-h^2)(2h) = \pi (72 h - 2 h^3) $$ A local extremum fulfills $$ 0 = V'(h) = \pi (72 - 6 h^2) \Rightarrow \\ 72 = 6 h^2 \Rightarrow \\ h = \pm \sqrt{12} = \pm 2 \sqrt{3} $$ Because $V''(h) < 0$ there we have a local maximum for $h = \pm 2 \sqrt{3}$.

The cylinder has a radius of $$ R = \sqrt{36 - h^2} = \sqrt{24} = 2 \sqrt{6} $$ and a height $$ H = 2h = 4 \sqrt{3} $$

Answer 2

using the angle $\theta$ and the radius $r$ of spherical cordinates we have the next two equalities: $$h=2r\cos\theta$$ and $$R=r\sin\theta$$ then the volume of the cylinder is: $$V=\pi hR^{2}=2\pi r\cos\theta r^{2}\sin^{2}\theta=2\pi r^{3}cos\theta\sin^{2}\theta$$ now we take the diferrential with respect $\theta$(Maximun criterion) $$\frac{dV}{d\theta}=-2\pi r^{3}\sin^{3}\theta+4\pi r^{3}\sin\theta\cos^{2}\theta=0$$ solve the equation for $theta$ we have: $$\theta=\arctan \sqrt{2}$$ put this result in the radius and hight formula, finally we have:

$$h=2r\cos(\arctan \sqrt{2})=2\cdot6\cdot\frac{1}{\sqrt{3}}$$ and $$R=r\sin(\arctan \sqrt{2})=6\cdot\sqrt{\frac{2}{3}}$$

We use the next identities: $$\cos\theta=\frac{1}{\sqrt{1+\tan^{2}\theta}}$$ and $$\sin\theta=\frac{1}{\sqrt{\frac{1}{\tan^{2}\theta}+1}}$$

Answer 3

So I quickly drew a cylinder inscribed in a sphere in MS paint for you, in case you needed more help setting up the problem (mvw's 3D plot is also very helpful for the intuition! Compare our pictures to get a feel for how everything goes together). I wasn't sure how to draw just part of the circles dotted as they technically should be, but this should help show what you need to do.

The cylinder itself is just the black part, the sphere is the red. I drew the height of the cylinder in green, the radius of the cylinder in purple, and the radius of the sphere (which you gave as 6 in) in orange.

As a general tip, when you are inscribing various shapes in circles/spheres, you should draw the radius of the circle/sphere where it intersects the inside object. You often create a right triangle with it.

We want to maximize the volume of the cylinder: $V = \pi r^2 h$.

We need to use our constraint that it is inscribed in a sphere of radius $R = 6$.

Try looking at the picture and writing a relationship between $r$ (radius of cylinder), $h$ (height of cylinder), and $R = 6$. See if you can continue from here!