Chapter 12.8, Problem 5CYU

To determine

To find: the surface area of the new pyramid.

Answer to Problem 5CYU

The surface area of the new pyramid is 450 square feet.

Explanation of Solution

Given:

A pyramid has a surface area of 50 square feet. The dimensions are tripled.

Calculation:

Since, the two pyramids are similar and a pair of solids is similar only if their corresponding measures are proportional.

Now, you know that the dimensions of the new pyramid are the triple of the original pyramid.

Hence, the scale factor of their side length is $\frac{1}{3}$ .

If two solids are similar with a scale factor of $\frac{a}{b}$ , then their surface areas have the ratio ${\left(\frac{a}{b}\right)}^2$ .

That is,

  $\frac{\text{Surface area of the original pyramid}}{\text{Surface area of the new pyramid}}={\left(\frac{a}{b}\right)}^2$

Let, the surface area of the new solid be S.

Now, substitute the known values in equation obtained above.

  $\begin{array}{l}\frac{50}{S}={\left(\frac{1}{3}\right)}^2\\ \Rightarrow \left(\frac{50}{S}\right)=\frac{1}{9}\end{array}$

Cross multiply, to obtain

  $\begin{array}{l}50\left(9\right)=1\left(S\right)\\ \Rightarrow S=450\end{array}$

Conclusion:

Therefore, the surface area of the new pyramid is 450 square feet.