Chapter 4, Problem 48RE

To determine

To find : The height and radius of the largest right circular cylinder.

Answer to Problem 48RE

The answer:

Height $h=2$

Radius $r=\sqrt{2}$

Explanation of Solution

Given information:

The figure is:

  

Calculation:

An angle. Half of the height of the cylinder is on the side of the triangle that is parallel to $h$ .

  

Find the volume equation using just one variable. Get another equation that connects $h$ and $r$ using the Pythagorean Theorem.

  $\begin{array}{c}V=\pi r^{2}h\\ $ \Rightarrow r^2+{\left(\frac{h}{2}\right)}^2={(\sqrt{3})}^2$\Rightarrow r^2+\frac{1}{4}{h}^2=3\\ \Rightarrow r^2=3-\frac{1}{4}{h}^2\\ V=\pi \left(3-\frac{1}{4}{h}^2\right)h\\ =\pi \left(3h-\frac{1}{4}{h}^3\right)\end{array}$

To find $V^{\prime }$ where it is $=0$

  $\begin{array}{c}{V}^{\prime }=\pi \left(3-\frac{3}{4}{h}^2\right)\\ \Rightarrow 0=3-\frac{3}{4}{h}^2\\ =12-3h^2\\ \Rightarrow 3h^2=12\\ \Rightarrow h^2=4\\ \Rightarrow h=2\end{array}$

To find $r$ , re-enter it into the second equation above.

  $\begin{array}{c}{r}^2=3-\frac{1}{4}{h}^2\\ $ =3-\frac{1}{4}{(2)}^2$=2\\ \Rightarrow r=\sqrt{2}\end{array}$

Therefore, the required height and radius of the largest right circular cylinder:

Height $h=2$

Radius $r=\sqrt{2}$