Chapter 12.7, Problem 25HP

Solution

To find: the scale factor.

Total scale factor is $4$

Given:

The total surface areaincreased by some scale factor is $24000m^2$

Formulas used:

The lateral area $L$ of rectangular prism is given by the formula

  $L=P.h$

The surface area $S$ of a rectangular prism is given by the formula

  $S=L+2B$

Calculation:

The objective is to find the scale factor.

Let $x$ be the scale factor. So the new dimension of the new rectangle are length is $90x$ and width is $40x$ and height is $90x$ .

The lateral area $L$ of rectangular prism is given by the formula

  $L=P.h$

Where $P$ is the perimeter of the base and $h$ is the height the rectangular prism

The surface area $S$ of a rectangular prism is given by the formula

  $S=L+2B$

Where $L$ is the lateral area and $B$ is the area of the base

First find the lateral area of the rectangular prism.

The perimeter $P$ of the prism is given by

  $\begin{array}{l}P=2(l+w)\\ \text{ =2(90x+40x)}\\ \text{ =2(130x)}\\ \text{ =260x}\end{array}$

Where $l$ is the length and $w$ is the width of the rectangular base

Put $\text{P=260x}$ and $\text{h=30}$ in the lateral area formula to find the lateral area of the prism

Thus

  $\begin{array}{l}\text{L=Ph}\\ \text{ =260x}\text{.30x}\\ {\text{ =7800x}}^2\end{array}$

Next find the surface area of the mew rectangular prism formed by multiplying the original dimension by $x$ .

First find the area $B$ of the base of the prism. The area $B$ of the base is given by

  $\begin{array}{l}B=l.w\\ \text{ =90x}\text{.40x}\\ {\text{ =3600x}}^2\end{array}$

Where $l$ is the length and $w$ is the width of the rectangular base

Put $B=3600x^2$ and $L=7800x^2$ in the surface area formula to find the surface area of the prism.

  $\begin{array}{l}S=L+2B\\ {\text{ =7800x}}^2+2.3600x^2\\ {\text{ =7800x}}^2+{\text{7200x}}^2\\ {\text{ =15000x}}^2\end{array}$

Since the surface area of the new rectangular prism is $240000m^2$ , SO

  $\begin{array}{l}{\text{15000x}}^2=240000\\ {\text{x}}^2=\frac{240000}{15000}\\ \text{ =16}\\ \text{x=}\pm \sqrt{16}\\ \text{ =}\pm 4\end{array}$ [Divide both sides by 1500]

Here $x\ne -4$ because the dimension is increased by the scale factor

Thus, the value of $x$ is $4$

Therefore, the scale factor is $4$ .