Exercise 43 Page 724

A sheet of paper with perimeter of $39$ inches must have the sum of its length and width equal $19.5.$ When creating a cylinder out of paper, one side of the paper will be the height and the other will be the circumference. We will repeat the process in Part A with various combinations of height and circumference to determine which size can be rolled into a right cylinder with the greatest volume. We saw in Part A that the cylinder with the greater circumference has the greater volume, so we know the circumference must be at least $11$ inches.

Height $h$	Circumference $C$	Radius found by $r=\frac{C}{2\pi }$	$V=\pi r^{2}h$
$7.5$	$12$	$1.91$	$85.9$
$7.25$	$12.25$	$1.95$	$86.5$
$7$	$12.5$	$1.99$	$87.0$
$6.75$	$12.75$	$2.03$	$87.3$
$6.5$	$13$	$2.07$	$87.4$
$6.25$	$13.25$	$2.11$	$87.3$
$6$	$13.5$	$2.15$	$87.0$

Looking at the table of values, we see that the volume is maximized when $h=6.5$ and $C=13.$ In other words, of all sheets of paper with perimeter $39$ inches (a $6.5$ inches by $13$ inches paper) can be rolled into a right cylinder with the greatest volume.