Chapter 12.7, Problem 30STP

To determine

To find:The scale factor of the dilation.

Answer to Problem 30STP

The scale factor of the dilation is 3

Explanation of Solution

Given:

Surface area of pyramid is 39 square feet.

Surface area is 351 square feet.

Calculation:

Consider the surface area of a square pyramid is $39f{t}^2$ and the surface area after dilation is $\text{351 }f{t}^2$ .

Let $a$ be the side of the base of the first square pyramid and $b$ be the side of the base of the square pyramid after dilation.

The scale factor of the dilation is $\frac{b}{a}$ .

Use the property that

  $\frac{\text{surface area of first square pyramid}}{\text{surface area of square pyramid after dilation}}={\left(\frac{a}{b}\right)}^2$

Thus,

  $\begin{array}{l}\text{351 }f{t}^2\\ \frac{\text{surface area of first square pyramid}}{\text{surface area of square pyramid after dilation}}={\left(\frac{a}{b}\right)}^2\\ \frac{\text{39}}{\text{351}}\text{=}{\left(\frac{\text{a}}{\text{b}}\right)}^{\text{2}}\\ \frac{\text{1}}{\text{9}}\text{=}{\left(\frac{\text{a}}{\text{b}}\right)}^{\text{2}}\\ \frac{\text{1}}{{\text{3}}^{\text{2}}}\text{=}{\left(\frac{\text{a}}{\text{b}}\right)}^{\text{2}}\end{array}$

  $\begin{array}{l}{\left(\frac{\text{a}}{\text{b}}\right)}^{\text{2}}=\frac{1}{3^2}\\ \frac{a}{b}=\sqrt{\frac{1}{3^2}}\\ \frac{a}{b}=\frac{1}{3}\\ \frac{b}{a}=\frac{3}{1}\end{array}$

Conclusion:Therefore, the scale factor of the dilation is 3.