Chapter 12.5, Problem 8WE

(a)

To determine

To find:The ratios of the heights of two similar pyramids.

Answer to Problem 8WE

The ratios of the heights of two similar pyramidsis $3:4$ .

Explanation of Solution

Given information:

The volumes of two similar pyramids are 3 and 375.

Calculation:

The ratio of the corresponding sides of any two similar geometric figures is termed as Scale factor.

Let the two volumes of pyramids be $V_1$ and $V_2$ , and scale factor here for both the pyramids be $h_1$ and $h_2$ .

The theorem of similar solids states that if the scale factor of two similar solids is $a:b$ , then the ratio of their volumes is $a^3:b^3$ .

Expression to determine the volumes of two similar pyramids from the above theorem is:

  $\begin{array}{c}\text{Volume}={\left(\text{ratio}\text{of}\text{heights}\right)}^3\\ \frac{V_1}{V_2}={\left(\frac{h_1}{h_2}\right)}^3\end{array}$

Substitute $3$ for $V_1$ and $375$ for $V_2$ in the above expression.

  $\begin{array}{c}\frac{3}{375}={\left(\frac{h_1}{h_2}\right)}^3\\ \frac{3÷3}{375÷3}={\left(\frac{h_1}{h_2}\right)}^3\\ {\left(\frac{h_1}{h_2}\right)}^3=\frac{1}{125}\end{array}$

To determine the ratio of heights, apply cube root at both sides in the above expression.

  $\begin{array}{c}\sqrt[3]{{\left(\frac{h_1}{h_2}\right)}^3}=\sqrt[3]{\frac{1}{125}}\\ \frac{h_1}{h_2}=\frac{1}{5}\end{array}$

Therefore, the ratios of heights of two similar pyramids is $1:5$ .

(b)

To determine

To find:The ratios of base areas of two similar pyramids.

Answer to Problem 8WE

The ratios of base areas of two similar pyramids is $1:25$ .

Explanation of Solution

Given information:

The volumes of two similar pyramids are $3$ and $375$ .

Calculation:

As calculated in part (a), the value of $h_1$ is $1$ and $h_2$ is $5$ .

The theorem of similar solids states that if the scale factor of two similar solids is $a:b$ , then the ratio of base areas, lateral areas and total areas is $a^2:b^2$ .

Expression to determine the area of two similar pyramids from the above theorem is:

  $\begin{array}{c}\text{base}\text{area}={\left(\text{side}\right)}^2={\left(\text{scale}\text{factor}\right)}^2\\ \frac{a_1}{a_2}={\left(\frac{h_1}{h_2}\right)}^2\end{array}$

To find the value of $\frac{a_1}{a_2}$ , substitute $1$ for $h_1$ and $5$ for $h_2$ in the above expression.

  $\begin{array}{c}\frac{a_1}{a_2}={\left(\frac{1}{5}\right)}^2\\ =\frac{1\times 1}{5\times 5}\\ =\frac{1}{25}\end{array}$

Therefore, the ratios of base areas of two similar pyramids is $1:25$ .

(c).

To determine

To find:The ratios of total areas of two similar pyramids.

Answer to Problem 8WE

The ratios of total areas of two similar pyramids is $1:25$ .

Explanation of Solution

Given information:

The volumes of two similar pyramids are $3$ and $375$ .

Calculation:

As calculated in part (a), the value of $h_1$ is $1$ and $h_2$ is $5$ .

Expression to determine the total areas of two similar pyramids from the above theorem is:

  $\begin{array}{c}\text{total}\text{area}={\left(\text{scale}\text{factor}\right)}^2\\ \frac{t.a_1}{t.a_2}={\left(\frac{h_1}{h_2}\right)}^2\end{array}$

Substitute $1$ for $h_1$ and $5$ for $h_2$ in the above expression.

  $\begin{array}{c}\frac{t.a_1}{t.a_2}={\left(\frac{1}{5}\right)}^2\\ =\frac{1\times 1}{5\times 5}\\ =\frac{1}{25}\end{array}$

Therefore, the ratios of total areas of two similar pyramids is $1:25$ .