Chapter 12.4, Problem 31WE

a.

To determine

To find: the volume of cylinder inscribed in the sphere with radius $10$ .

Answer to Problem 31WE

The volume of inscribed cylinder is $2\pi x\left(100-x^2\right)$ .

Explanation of Solution

Given: A cylinder inscribed in the sphere with radius $10$ and height of cylinder is $2x$

  

Calculation:

Calculate the radius of base of inscribed cylinder say $r$ .

  $\begin{array}{l}r=\sqrt{{10}^2-x^2}\\ r=\sqrt{100-x^2}\end{array}$

And

Calculate the volume of inscribed cylinder.

  $\begin{array}{l}V=\pi r^{2}h\\ =\pi {\left(\sqrt{100-x^2}\right)}^2\left(2x\right)\\ =2\pi x\left(100-x^2\right)\end{array}$

Therefore, the volume of inscribed cylinder is $2\pi x\left(100-x^2\right)$ .

b.

To determine

To find: the maximum volume of cylinder by substituting the value of $x$ .

Answer to Problem 31WE

The maximum volume of inscribed cylinder is $\frac{4000\sqrt{3}\pi }{9}{\text{ unit}}^3$ .

Explanation of Solution

Given:

Value of $x$ to substitute:

  $x=\frac{10\sqrt{3}}{3}$

Calculation:

Calculate the volume of cylinder by substituting $x$ .

  $\begin{array}{l}V=2\pi x\left(100-x^2\right)\\ =2\pi \left(\frac{10\sqrt{3}}{3}\right)\left(100-{\left(\frac{10\sqrt{3}}{3}\right)}^2\right)\\ =2\pi \left(\frac{10\sqrt{3}}{3}\right)\left(100-\frac{300}{9}\right)\\ =2\pi \left(\frac{10\sqrt{3}}{3}\right)\left(100-\frac{100}{3}\right)\end{array}$

  $\begin{array}{l}=2\pi \left(\frac{10\sqrt{3}}{3}\right)\left(\frac{300-100}{3}\right)\\ =2\pi \left(\frac{10\sqrt{3}}{3}\right)\left(\frac{200}{3}\right)\\ =\frac{4000\sqrt{3}\pi }{9}{\text{ unit}}^3\end{array}$

Therefore, the maximum volume of inscribed cylinder is $\frac{4000\sqrt{3}\pi }{9}{\text{ unit}}^3$ .