Chapter 12.7, Problem 9P

To determine

To calculate: How many times greater is the radius of a cylinder 3 m high that the radius of a cylinder 6 m high with the same volume with the same volume, given the height of a circular cylinder of the given volumes varies inversely as the square of the radius of the base.

Answer to Problem 9P

The radius of the 3 m high cylinder is twice the radius of 6 m high cylinder.

  $r_1^2=2\cdot r_2^2{}^{}$

Explanation of Solution

Given information:

The height of a circular cylinder of the given volumes varies inversely as the square of the radius of the base.

Let, $r_1^{}$ radius of the cylinder of height 3m.

  $r_2^{}$ radius of the cylinder of height 6 m.

And $h\prec \frac{1}{r^2}$

Let k be a constant

  $h=\frac{k}{r{}^2}$

Height of the cylinder 1 is

  $h_1^{}=\frac{k}{r_1^2{}^{}}$

  $3=\frac{h}{r_1^2}$...... (1)

Height of the cylinder 2 is

  $h_2^{}=\frac{k}{r_1^2}$

  $6=\frac{k}{r_2^2}$...... (2)

Now divide equation 1 and 2.

  $\begin{array}{l}\frac{3}{6}=\frac{r_2^2}{r_1^2}\\ \frac{1}{2}=\frac{r_2^2}{r_1^2}\\ r_1^2=2\cdot r_2^2{}^{}\end{array}$