Chapter 4.3, Problem 13CFU

To determine

To find: The radius and the height to the nearest inch if the volume of the cylinder is $5\pi$ cubic inches.

Answer to Problem 13CFU

  $\text{Radius}=1\text{ in, Height}=5\text{ in}\text{.}$

Explanation of Solution

Given information: Geometry: A cylinder has a height 4 inches greater than the radius of its base.

Calculation:

Using the given data and a formula for volume, we arrive at an equation,

  $\begin{array}{l}h=r+4V=\pi r^{2}h\\ V=\pi r^2\left(r+4\right)\\ 5\pi =\pi r^2\left(r+4\right)\end{array}$

This involves a polynomial of degree 3.

If we find one zero, we will find one solution to the equation,

  $\begin{array}{l}5\pi =\pi r^3+4\pi r^2\\ 0=\pi r^3+4\pi r^2-5\pi \\ 0=\pi \left(r^3+4r^2-5\right)\end{array}$

Before any serious investigation, we try $r=1,-1$ etc to see if,

One root is r = 1 in.

  $\begin{array}{l}h=r+4\\ h=1+4=5\text{ in}\end{array}$

Thus,

  $\text{Radius}=1\text{ in, Height}=5\text{ in}\text{.}$