Chapter 12.5, Problem 5CE

(a).

To determine

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

Answer to Problem 5CE

The ratios of volumes of two similar sphere is $1:8$ .

Explanation of Solution

Given information:

The volumes of two similar spheres are $2\pi {\text{m}}^{\text{3}}$ and $16\pi {\text{m}}^{\text{3}}$ respectively.

Calculation:

Let, the two volumes be $V_1$ and $V_2$ .

Expression of given two volumes of similar spheres in ratio is:

  $\frac{V_1}{V_2}=\frac{2\pi {\text{m}}^{\text{3}}}{16\pi {\text{m}}^{\text{3}}}$

To find the ratio of volumes, divide the numerator and denominator of the above expression by $2$ .

  $\begin{array}{c}\frac{V_1}{V_2}=\frac{2\pi {\text{m}}^{\text{3}}÷2}{16\pi {\text{m}}^{\text{3}}÷2}\\ =\frac{1}{8}\end{array}$

Therefore, the volume ratio of two similar spheres in reduced form is $1:8$ .

(b).

To determine

To find:The ratios of diameters of two similar sphere.

Answer to Problem 5CE

The ratios of diameters of two similar sphere is $1:2$ .

Explanation of Solution

Given information:

The volumes of two similar spheres are $2\pi {\text{m}}^{\text{3}}$ and $16\pi {\text{m}}^{\text{3}}$ respectively.

Calculation:

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

As calculated in part(a), The value of $V_1$ is $1$ and $V_2$ is $8$ .

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 diameters from volumes of spheresusing the above theorem is:

  $\frac{V_1}{V_2}={\left(\frac{d_1}{d_2}\right)}^3$

Substitute $1$ for $V_1$ and $8$ for $V_2$ in the above expression.

  $\frac{1}{8}={\left(\frac{d_1}{d_2}\right)}^3$

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

  $\begin{array}{c}\sqrt[3]{\frac{1}{8}}=\sqrt[3]{{\left(\frac{d_1}{d_2}\right)}^3}\\ \frac{d_1}{d_2}=\frac{1}{2}\end{array}$

Therefore, the ratio of diameter of two similar sphere is $1:2$ .

(c).

To determine

To find:The ratios of areas of two similar spheres.

Answer to Problem 5CE

The ratios of areas of two similar spheres is $1:4$ .

Explanation of Solution

Given information:

The volumes of two similar spheres are $2\pi {\text{m}}^{\text{3}}$ and $16\pi {\text{m}}^{\text{3}}$ respectively.

Calculation:

As calculated in part(b), The value of $d_1$ is $1$ and $d_2$ is $2$ .

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 theratio of area of spheresfrom 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 $1$ for $d_1$ and $2$ for $d_2$ in the above expression.

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

Therefore, the ratios of areas of two similar spheres is $1:4$ .