Chapter 12.5, Problem 5WE

(a)

To determine

To find:The ratios of lengths of Earth and moon equators.

Answer to Problem 5WE

The ratios of equators length of Earth and moon is $4:1$ .

Explanation of Solution

Given information:

The diameter of Earth is $12,800\text{km}$ and the diameter of moon is $3,200\text{km}$ .

Calculation:

The ratio of the corresponding sides of any two similar geometric figures is termed as Scale factor. Let, the diameters of Earth and moon be $d_1$ and $d_2$ .

The length of the equator of earth and moon is equal to diameter of the Earth and moon.

To find the equators lengths, write the corresponding diameter in the ratio equating to length ratio.

  $\frac{\text{equator}{\text{length}}_{\text{earth}}}{\text{equator}{\text{length}}_{\text{moon}}}=\frac{d_1}{d_2}$

Substitute $12,800\text{km}$ for $d_1$ and $3,200\text{km}$ for $d_2$ in the above expression.

  $\begin{array}{c}\frac{\text{equator}{\text{length}}_{\text{earth}}}{\text{equator}{\text{length}}_{\text{moon}}}=\frac{12800}{3200}\\ =\frac{12800÷3200}{3200÷3200}\\ =\frac{4}{1}\end{array}$

Therefore, the equators length ratios of Earth and moon is $4:1$ .

(b)

To determine

To find:The ratios of areas of Earth and Moon.

Answer to Problem 5WE

The ratios of areas of Earth and Moon is $16:1$ .

Explanation of Solution

Given information:

The diameter of Earth is $12,800\text{km}$ and the diameter of moon is $3,200\text{km}$ .

Calculation:

As calculated in part(a), the value of $d_1$ equals to $\text{equator}{\text{length}}_{\text{earth}}$ is $4$ and $d_2$ equals to $\text{equator}{\text{length}}_{\text{moon}}$ is $1$ .

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

Expression to determine the area of Earth and moon from the above theorem is:

  $\begin{array}{c}\text{area}={\left(\text{scale}\text{factor}\right)}^2\\ \frac{a_1}{a_2}={\left(\frac{d_1}{d_2}\right)}^2\end{array}$

Substitute $4$ for $d_1$ and $1$ for $d_2$ in the above expression.

  $\begin{array}{c}\frac{a_1}{a_2}={\left(\frac{4}{1}\right)}^2\\ =\frac{4\times 4}{1\times 1}\\ =\frac{16}{1}\end{array}$

Therefore, the ratios of areas of Earth and Moon is $16:1$ .

(c)

To determine

To find:The ratios of volumes ofEarth and Moon.

Answer to Problem 5WE

The ratios of volumes of Earth and Moon is $64:1$ .

Explanation of Solution

Given information:

The diameter of Earth is $12,800\text{km}$ and the diameter of moon is $3,200\text{km}$ .

Calculation:

As calculated in part(a), the value of $d_1$ equals to $\text{equator}{\text{length}}_{\text{earth}}$ is $4$ and $d_2$ equals to $\text{equator}{\text{length}}_{\text{moon}}$ is $1$ .

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 ratio of volumes of Earth and moon from the above theorem is:

  $\begin{array}{c}\text{Volumes}={\left(\text{scale}\text{factor}\right)}^3\\ \frac{V_1}{V_2}={\left(\frac{d_1}{d_2}\right)}^3\end{array}$

Substitute $4$ for $d_1$ and $1$ for $d_2$ in the above expression.

  $\begin{array}{c}\frac{V_1}{V_2}={\left(\frac{4}{1}\right)}^3\\ =\frac{4\times 4\times 4}{1\times 1\times 1}\\ =\frac{64}{1}\end{array}$

Therefore, the ratios of volumes of Earth and Moon is $64:1$ .