Chapter 12.2, Problem 27PPS

(a)

To determine

To find: the volume of the figure

Answer to Problem 27PPS

The volume of the rectangular box is $42{\text{ cm}}^3$ .

Explanation of Solution

Given:

Consider the figure

  

Formula used:

The volume V of the rectangular box is $V=l.w.h$

Calculation:

The objective is to find the volume of the figure.

The volume V of the rectangular box is given by the formula

  $V=l.w.h$

Where l is the length, w is the width and h is the height of the rectangular box.

Put $l=7$ , $w=2$ and $h=8$ in the volume formula to find the volume.

Thus, the volume V is

  $\begin{array}{l}V=l.w.h\\ =7\cdot 2\cdot 3\\ =42\end{array}$

Conclusion:

Therefore, the volume of the rectangular box is $42{\text{ cm}}^3$ .

(b)

To determine

To explain:the volume change if one of the dimensions is doubled.

Answer to Problem 27PPS

Therefore, when three of the dimensions are doubled then the new volume is also 8 times the old volume.

Explanation of Solution

Calculation:

The objective is to find the change in volume when one of the dimensions is doubled.

Let length l is doubled. Let the new volume is $V_1$ .

Thus, the new volume $V_1$ is

  $\begin{array}{l}{V}_1=2l.w.h\\ =2\left(l.w.h\right)\\ =2V\end{array}$

When one of the dimensions is doubled then the new volume is also doubled.

The objective is to find the change in volume when two of the dimensions are doubled.

Let l length and width w is doubled. Let the new volume is $V_2$

Thus, the new volume $V_2$ is

  $\begin{array}{l}{V}_2=2l.2w.h\\ =4\left(l.w.h\right)\\ =4V\end{array}$

When two of the dimensions are doubled then the new volume is also 4 times the old volume.

The objective is to find the change in volume when three of the dimensions are doubled.

Let length l , width w and height h is doubled. Let the new volume is $V_3$ .

Thus, the new volume $V_3$ is

  $\begin{array}{l}{V}_3=2l.2w.2h\\ =8\left(l.w.h\right)\\ =8V\end{array}$

When three of the dimensions are doubled then the new volume is also 8 times the old volume

Conclusion:

Therefore, when three of the dimensions are doubled then the new volume is also 8 times the old volume.

(c)

To determine

To discuss: the repeat above steps and triple each dimension.

Answer to Problem 27PPS

When three of the dimensions are triple then the new volume is also 27 times the old volume.

Explanation of Solution

Calculation:

The objective is to find the change in volume when one of the dimensions is triple.

Let lengthl is triple. Let the new volume is $V_1$ .

Thus, the new volume $V_1$ is

  $\begin{array}{l}{V}_1=3l.w.h\\ =3\left(l.w.h\right)\\ =3V\end{array}$

When one of the dimensions is triple then the new volume is also triple.

The objective is to find the change in volume when two of the dimensions are triple.

Let length l and width w is triple. Let the new volume is $V_2$ .

Thus, the new volume $V_2$ is

  $\begin{array}{l}{V}_2=3l.3w.h\\ =9\left(l.w.h\right)\\ =9V\end{array}$

When two of the dimensions are triple then the new volume is also 9 times the old volume

The objective is to find the change in volume when three of the dimensions are triple.

Let length l width w and heighth his triple. Let the new volume is $V_3$ .

Thus, the new volume $V_3$ is

  $\begin{array}{l}{V}_3=3l.3w.3h\\ =27\left(l.w.h\right)\\ =27V\end{array}$

When three of the dimensions are triple then the new volume is also 27 times the old volume.

Conclusion:

Therefore,when three of the dimensions are triple then the new volume is also 27 times the old volume.

(d)

To determine

To discuss: the repeat above steps and triple each dimension.

Answer to Problem 27PPS

When each of the dimensions is multiplied by 6 then the new volume is also 216 times the old volume.

Explanation of Solution

Calculation:

The objective is to find the change in volume when each of the dimensions are multiplied by 6. Let length l , width w and height h his multiplied by 6. Let the new volume is $V_1$ .

Thus, the new volume $V_1$ is

  $\begin{array}{l}{V}_1=6^{3}V\\ =216V\end{array}$

When each of the dimensions is multiplied by 6 then the new volume is also 216 times the old volume.

Conclusion:

Therefore,when each of the dimensions is multiplied by 6 then the new volume is also 216 times the old volume.