Chapter 12.8, Problem 6CYU

(a)

To determine

To find: the scale factor between the prototype and the actual box.

Answer to Problem 6CYU

The scale factor between the porotype and the actual box is $\frac{5}{14}$ .

Explanation of Solution

Given:

Theheight of the new box is 10 cm

Length of the new box is 3 cm

Width of the new box is 2 cm.

The height of the actual box is 28 cm.

Calculation:

The objective is to find the scale factor between the prototype and the actual box.

Two solid figures are similar if they have the same shape and their corresponding linear measurements are proportional.

Since the new box is prototype of actual box, so the boxes are similar.

The ratio of the height new box to the height of actual box is $\frac{10}{28}$ that is $\frac{5}{14}$ .

Conclusion:

Therefore, the scale factor between the prototype and the actual box is $\frac{5}{14}$ .

(b)

To determine

To find:the width of the actual box.

Answer to Problem 6CYU

The width of the actual box is 5.6 cm.

Explanation of Solution

Calculation:

The objective is to find the width of the actual box.

Let x be the width of the actual box.

Since the boxes are similar, so their corresponding linear measurements are proportional.

Thus,

  $\begin{array}{l}\frac{10}{28}=\frac{2}{x}\\ 10x=28\cdot 2[CrossMultiply]\\ =56\\ x=\frac{56}{10}\text{ [Divide both sides by 10]}\\ \text{= 5}\text{.6}\end{array}$

Conclusion:

Therefore, the width of the actual box is 5.6 cm.

(c)

To determine

To find: volume of the actual box.

Answer to Problem 6CYU

The volume of the actual box is $\text{1317}{\text{.12 cm}}^3$ .

Explanation of Solution

Calculation:

The objective is to find the volume of the actual box.

First find the volume of the new box.

The volume V of the new box is given by

  $V=l\cdot w\cdot h$ , where l is the length, w is the width and h is the height of the box.

  $\begin{array}{l}=3\cdot 2\cdot 10\\ =60\end{array}$

Thus, the volume of the new box is $60c{m}^3$ .

To find the volume of the actual box use the property that if two solids are similar with scale

factor of $\frac{a}{b}$ then the volume are in the ratio ${\left(\frac{a}{b}\right)}^3$ .

Since two boxes are similar, so

  $\begin{array}{l}\frac{\text{Volume of new box}}{\text{Volume of actual box}}={\left(\frac{a}{b}\right)}^3,\text{ where }\frac{a}{b}\text{ is the scale factor}\text{.}\\ \frac{60}{\text{Volume of actual box}}={\left(\frac{5}{14}\right)}^3\\ \text{ = }\frac{5^3}{{14}^3}\\ \text{ = }\frac{125}{2744}\\ 60\cdot 2744=125\cdot \text{Volume of actual box}\\ \text{164640 = 125}\cdot \text{ Volume of actual box}\\ \text{125}\cdot \text{ Volume of actual box}=\text{ 164640}\\ \text{Volume of actual box}=\frac{\text{164640}}{125}[Dividebothsides\text{ by 125}]\\ =\text{ 1317}\text{.12}\end{array}$

Conclusion:

Therefore, the volume of the actual box is $\text{1317}{\text{.12 cm}}^3$ .