Chapter 12.8, Problem 2CYP

To determine

Find the ratio of the height of the large prism to the height of the small prism.

Answer to Problem 2CYP

7:3

Explanation of Solution

Given:

Two similar prisms have surface areas of 98 square centimeters and 18 square centimeters.

Calculation:

Since the two prisms are similar, so, let the ratio of corresponding sides of large to small prism is equal to k .

Ratio of the perimeter of base is $\frac{P_{l\arg e}}{P_{small}}=\frac{\text{sum of sides of base of large prism}}{\text{sum of sides of base of small prism}}=k\Rightarrow P_{l\arg e}=k{P}_{small}$ .

Now, the ratio of surface areas :

  $\begin{array}{l}\frac{S{A}_{l\arg e}}{S{A}_{small}}\\ =\frac{P_{l\arg e}{h}_{l\arg e}}{P_{small}{h}_{small}}\\ =\frac{k{P}_{small}{h}_{l\arg e}}{P_{small}{h}_{small}}\\ =k\frac{h_{l\arg e}}{h_{small}}\\ =k^2\end{array}$

Also,

  $\frac{S{A}_{l\arg e}}{S{A}_{small}}=\frac{98}{18}=\frac{49}{9}$

Hence,

  $\begin{array}{l}\frac{S{A}_{l\arg e}}{S{A}_{small}}=\frac{49}{9}=k^2\\ \Rightarrow k^2=\frac{49}{9}\\ \Rightarrow k=\sqrt{\frac{49}{9}}\mathrm{.................}[\text{take square root each side]}\\ \Rightarrow k=\frac{7}{3}\\ \Rightarrow \frac{h_{l\arg e}}{h_{small}}=\frac{7}{3}\end{array}$

So, the ratio of the height of the large prism to the height of the small prism is 7:3.