Chapter 12.2, Problem 18PPS

To determine

To find: the volume of the foam pieces.

Answer to Problem 18PPS

The volume of the foam pieces needed to fill the space between the candle and the box is $85.5{\text{ in}}^3$ .

Explanation of Solution

Given:

  

Formula used:

The volume V of the triangular prism is $V=Bh$

Calculation:

The objective is to find the volume of the foam pieces needed to fill the space between the candle and the box.

The volume V of a triangular prism is given by the formula

  $V=Bh$

Where B is the area of a face and h is the height i.e. length of the prism.

Find the volume $V_1$ of one of the triangular prism.

From the figure the area B is a triangle whose base is 3 in. and height 2.6 in.

The area B of the triangle is given by

  $\begin{array}{l}B=\frac{1}{2}\cdot b\cdot a\text{ Where }b\text{ is the base and }a\text{ is the height of the triangle}\\ \text{=}\frac{1}{2}\cdot 13\left(2.6\right)\\ =3\left(1.3\right)\\ =3.9\end{array}$

Put $B=3.9$ and $h=5$ in the volume formula to find the volume of the prism.

Thus, the volume $V_1$ is

  $\begin{array}{l}{V}_1=Bh\\ =\left(3.9\right)5\\ =19.5\end{array}$

Therefore, the volume of the triangular prism is $19.5{\text{ in}}^3$ .

Now, find the volume of the rectangular box

The volume V of a rectangular box is given by

  $V=l.w.h$

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

The length of the rectangular box is $l=5$ , width of the rectangular box is $w=3$ and height of the rectangular box is $h=7$ .

The volume $V_2$ of the rectangular box is given by

  $\begin{array}{l}{V}_2=l.w.h\\ =5\cdot 3\cdot 7\\ =105\end{array}$

Therefore, the volume of the rectangular box is $105{\text{ in}}^3$ .

Thus, the volume V of the foam pieces needed to fill the space between the candle and the box is

  $\begin{array}{l}V=V_2-V_1\\ =105-19.5\\ =85.5\end{array}$

Conclusion:

Therefore, the volume of the foam pieces needed to fill the space between the candle and the box is $85.5{\text{ in}}^3$ .