Chapter 6.5, Problem 17E

To determine

To calculate: The dimensions of three similar boxes.

Answer to Problem 17E

The first similar figure has a length of $\text{6 inches}$ , width of $\text{4 inches}$ and height of $\text{8 inches}$ .

The second similar figure has a length of $\text{9 inches}$ , width of $\text{6 inches}$ and height of $\text{12 inches}$ .

The third similar figure has a length of $\text{4}\text{.5 inches}$ , width of $\text{3 inches}$ and height of $\text{6 inches}$ .

Explanation of Solution

Given information:

Length of the rectangular box = $3\text{ inches}$

Width of the rectangular box = $\text{2 inches}$

Height of the rectangular box = $\text{4 inches}$

Length of the first similar box = $\text{6 inches}$

Width of the second similar box = $\text{6 inches}$

Height of the third similar box = $\text{6 inches}$

Calculation:

To find dimensions of the similar boxes, use properties of similar figures. The ratio of lengths of corresponding sides of similar figures are equal.

For the first similar figure:

Let width be $w_1$ . Width can be calculated as:

  $\begin{array}{l}\frac{\text{Length of rectangle}}{\text{Length of first similar figure}}=\frac{\text{Width of rectangle}}{\text{Width of first similar figure}}\text{ [Write proportion]}\\ \frac{3}{6}=\frac{2}{w_1}\text{ [Substitute values]}\\ \text{ 3}\cdot w_1=2\cdot 6\text{ [Form cross product]}\\ \text{ 3}{w}_1=12\text{ [Divide both sides by 3]}\\ w_1=4\text{ [Simplify]}\end{array}$

Let height be $h_1$ . Height can be calculated as:

  $\begin{array}{l}\frac{\text{Length of rectangle}}{\text{Length of first similar figure}}=\frac{\text{Height of rectangle}}{\text{Height of first similar figure}}\text{ [Write proportion]}\\ \frac{3}{6}=\frac{4}{h_1}\text{ [Substitute values]}\\ \text{ 3}\cdot h_1=4\cdot 6\text{ [Form cross product]}\\ \text{ 3}{h}_1=24\text{ [Divide both sides by 3]}\\ h_1=8\text{ [Simplify]}\end{array}$

Therefore, the first similar figure has a length of $\text{6 inches}$ , width of $\text{4 inches}$ and height of $\text{8 inches}$ .

For the second similar figure:

Let length be $l_2$ . length can be calculated as:

  $\begin{array}{l}\frac{\text{Width of rectangle}}{\text{Width of second similar figure}}=\frac{\text{Length of rectangle}}{\text{Length of second similar figure}}\text{ [Write proportion]}\\ \frac{2}{6}=\frac{3}{l_2}\text{ [Substitute values]}\\ \text{ 2}\cdot l_2=3\cdot 6\text{ [Form cross product]}\\ \text{ 2}{l}_2=18\text{ [Divide both sides by 2]}\\ l_2=9\text{ [Simplify]}\end{array}$

Let height be $h_2$ . Height can be calculated as:

  $\begin{array}{l}\frac{\text{Width of rectangle}}{\text{Width of second similar figure}}=\frac{\text{Height of rectangle}}{\text{Height of second similar figure}}\text{ [Write proportion]}\\ \frac{2}{6}=\frac{4}{h_2}\text{ [Substitute values]}\\ \text{ 2}\cdot h_2=4\cdot 6\text{ [Form cross product]}\\ \text{ 2}{h}_2=24\text{ [Divide both sides by 2]}\\ h_2=12\text{ [Simplify]}\end{array}$

Therefore, the second similar figure has a length of $\text{9 inches}$ , width of $\text{6 inches}$ and height of $\text{12 inches}$ .

For the third similar figure:

Let length be $l_3$ . length can be calculated as:

  $\begin{array}{l}\frac{\text{Height of rectangle}}{\text{Height of third similar figure}}=\frac{\text{Length of rectangle}}{\text{Length of third similar figure}}\text{ [Write proportion]}\\ \frac{4}{6}=\frac{3}{l_3}\text{ [Substitute values]}\\ \text{ 4}\cdot l_3=3\cdot 6\text{ [Form cross product]}\\ \text{ 4}{l}_3=18\text{ [Divide both sides by 4]}\\ l_3=4.5\text{ [Simplify]}\end{array}$

Let width be $w_3$ . Width can be calculated as:

  $\begin{array}{l}\frac{\text{Height of rectangle}}{\text{Height of third similar figure}}=\frac{\text{Width of rectangle}}{\text{Width of third similar figure}}\text{ [Write proportion]}\\ \frac{4}{6}=\frac{2}{w_3}\text{ [Substitute values]}\\ \text{ 4}\cdot w_3=2\cdot 6\text{ [Form cross product]}\\ {\text{ 4w}}_3=12\text{ [Divide both sides by 4]}\\ {\text{ w}}_3=3\text{ [Simplify]}\end{array}$

Therefore, the third similar figure has a length of $\text{4}\text{.5 inches}$ , width of $\text{3 inches}$ and height of $\text{6 inches}$ .