Chapter 12.5, Problem 4WE

(a)

To determine

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

Answer to Problem 4WE

The ratios of base areas of two similar pyramids is $4:9$ .

Explanation of Solution

Given information:

The heights of two similar pyramids are $12$ and $18$ .

Calculation:

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

Let the two heights of pyramids be $h_1$ and $h_2$ , and scale factor for both the pyramids be $s_1$ and $s_2$ .

To find the scale factor, write the corresponding heights in the ratio equating to scale factor.

  $\frac{s_1}{s_2}=\frac{h_1}{h_2}$

Substitute $12$ for $h_1$ and $18$ for $h_2$ in the above expression.

  $\begin{array}{c}\frac{s_1}{s_2}=\frac{12}{18}\\ =\frac{12÷6}{18÷6}\\ =\frac{2}{3}\end{array}$

Therefore, from the above expression the value of $s_1$ is $2$ and $s_2$ is $3$ .

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:

  $\begin{array}{c}\text{base}\text{area}={\left(\text{scale}\text{factor}\right)}^2\\ \frac{a_1}{a_2}={\left(\frac{s_1}{s_2}\right)}^2\end{array}$

To find the value of $\frac{a_1}{a_2}$ , substitute $2$ for $s_1$ and $3$ for $s_2$ in the above expression.

  $\begin{array}{c}\frac{a_1}{a_2}={\left(\frac{2}{3}\right)}^2\\ =\frac{2\times 2}{3\times 3}\\ =\frac{4}{9}\end{array}$

Therefore, the ratios of base areas of two similar pyramids is $4:9$ .

(b).

To determine

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

Answer to Problem 4WE

The ratios of lateral areas of two similar pyramids is $4:9$ .

Explanation of Solution

Given information:

The heights of two similar pyramids are $12$ and $18$ .

Calculation:

As calculated in part(a), the value of $s_1$ is $2$ and $s_2$ is $3$ .

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

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

  $\begin{array}{c}\text{lateral}\text{area}={\left(\text{scale}\text{factor}\right)}^2\\ \frac{l.a_1}{l.a_2}={\left(\frac{s_1}{s_2}\right)}^2\end{array}$

Substitute $2$ for $s_1$ and $3$ for $s_2$ in the above expression.

  $\begin{array}{c}\frac{l.a_1}{l.a_2}={\left(\frac{2}{3}\right)}^2\\ =\frac{2\times 2}{3\times 3}\\ =\frac{4}{9}\end{array}$

Therefore, the ratios of lateral areas of two similar pyramids is $4:9$ .

(c).

To determine

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

Answer to Problem 4WE

The ratios of total areas of two similar pyramids is $4:9$ .

Explanation of Solution

Given information:

The heights of two similar pyramids are $12$ and $18$ .

Calculation:

As calculated in part(a), the value of $s_1$ is $2$ and $s_2$ is $3$ .

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

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

  $\begin{array}{c}\text{total}\text{area}={\left(\text{scale}\text{factor}\right)}^2\\ \frac{t.a_1}{t.a_2}={\left(\frac{s_1}{s_2}\right)}^2\end{array}$

Substitute $2$ for $s_1$ and $3$ for $s_2$ in the above expression.

  $\begin{array}{c}\frac{t.a_1}{t.a_2}={\left(\frac{2}{3}\right)}^2\\ =\frac{2\times 2}{3\times 3}\\ =\frac{4}{9}\end{array}$

Therefore, the ratios of total areas of two similar pyramids is $4:9$ .

(d).

To determine

To find:The ratios of volumes of two similar pyramids.

Answer to Problem 4WE

The ratios of volumes of two similar pyramids is $4:9$ .

Explanation of Solution

Given information:

The heights of two similar pyramids are $12$ and $18$ .

Calculation:

As calculated in part(a), the value of $s_1$ is $2$ and $s_2$ is $3$ .

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:

  $\begin{array}{c}\text{Volumes}={\left(\text{scale}\text{factor}\right)}^3\\ \frac{V_1}{V_2}={\left(\frac{s_1}{s_2}\right)}^3\end{array}$

Substitute $2$ for $s_1$ and $3$ for $s_2$ in the above expression.

  $\begin{array}{c}\frac{V_1}{V_2}={\left(\frac{2}{3}\right)}^3\\ =\frac{2\times 2\times 2}{3\times 3\times 3}\\ =\frac{8}{27}\end{array}$

Therefore, the ratios of volumes of two similar pyramids is $8:27$ .