Chapter 9.1, Problem 54PPS

a.

To determine

An equation showing the relationship between side length s and volume v of a cube.

Answer to Problem 54PPS

Volume of cube $v=s^3$ .

Explanation of Solution

Given information:

The volume of a cube equals the side length cubed.

Calculation:

Volume of cube = (side length)3

$v=s^3$

b.

To determine

A table of values showing the volume of a cube with side length of 1,2,4,8,and 16 centimeters.

Answer to Problem 54PPS

The required answer is

  $\begin{array}{l}{1}^3=1c{m}^3\\ 2^3=8c{m}^3\\ 4^3=64c{m}^3\\ 8^3=512c{m}^3\\ {16}^3=4096c{m}^3\end{array}$

Explanation of Solution

Given information:

The volume of a cube equals the side length cubed.

Calculation:

Side length	volume
$1cm$	$1^3=1c{m}^3$   $\left(1^3=1\cdot 1\cdot 1=1\right)$
$2cm$	$2^3=8c{m}^3$   $\left(2^3=2\cdot 2\cdot 2=8\right)$
$4cm$	$4^3=64$   $\left(4^3=4\cdot 4\cdot 4=64\right)$
$8cm$	$8^3=512c{m}^3$   $\left(8^3=8\cdot 8\cdot 8=512\right)$
$16cm$	${16}^3=4096$   $\left({16}^3=16\cdot 16\cdot 16=4096\right)$

c.

To determine

The change in volume when the side length of a cube is doubled.

Answer to Problem 54PPS

The required answer is volume $=8v$

Explanation of Solution

Given information:

The volume of a cube equals the side length cubed.

Calculation:

Side length	New length	Volume
$1cm$	$2\cdot 1=2cm$	$2^3=8$   $\left(2^3=2\cdot 2\cdot 2=8\right)$
$2cm$	$2\cdot 2=4cm$	$4^3=64$   $\left(4^3=4\cdot 4\cdot 4=64\right)$
$4cm$	$2\cdot 4=8cm$	$8^3=512$   $\left(8^3=8\cdot 8\cdot 8=512\right)$
$8cm$	$2\cdot 8=16cm$	${16}^3=4096$   $\left({16}^3=16\cdot 16\cdot 16=4096\right)$
$16cm$	$2\cdot 16=32cm$	${32}^3=32768$   $\left({32}^3=32\cdot 32\cdot 32=32768\right)$

Algebraic expression for double side length:

Original length $=s$

Original volume $=v=s^3$

New length $=2s$

New volume $=v^{\text{'}}={\left(2s\right)}^3=8s^3$

Therefore , for double side length, $2s$ ,

Volume $=8\cdot \left(originalvolume\right)$

$=8v$