Chapter 8.1, Problem 43PPSE

Solution

To state: At least four ways to change the volume and radius of a cylinder so that its height is quadrupled if height h of a cylinder varies directly with its volume V and inversely with the square of its radius r .

The change in volume and radius results in change in height.

Given information:

The height h of a cylinder varies directly with its volume V and inversely with the square of its radius r .

Explanation:

The height h of a cylinder varies directly with its volume V and inversely with the square of its radius r .

Then it can be written as: $h=\frac{kV}{r^2}$

Now replace $V\text{by}4V$ ,

  $h=\frac{k\left(4V\right)}{r^2}$

The height is 4 times of the original height. That is if the volume of the cylinder is 4 times of the original volume then its height is quadrupled.

Now replace $V\text{by}16V\text{and}r\text{by}2r$ ,

  $\begin{array}{c}h=\frac{16kV}{{\left(2r\right)}^2}\\ =\frac{4kV}{r^2}\end{array}$

The height is 4 times of the original height. That is if the volume of the cylinder is 16 times and radius is 2 times of the original values then its height is quadrupled.

Now replace $r\text{by}\frac{r}{2}$ ,

  $\begin{array}{c}h=\frac{kV}{{\left(\frac{r}{2}\right)}^2}\\ =\frac{4kV}{r^2}\end{array}$

The height is 4 times of the original height.

That is if the radius of the cylinder is half of the original radius then its height is quadrupled.