Chapter 12.5, Problem 7WE

(a).

To determine

To find:The ratios of the radii of two similar cones.

Answer to Problem 7WE

The ratios of the radii of two similar conesis $2:3$ .

Explanation of Solution

Given information:

The volumes of two similar cones are $8\pi$ and $27\pi$ .

Calculation:

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

Let the two Volumes of cones be $V_1$ and $V_2$ and scale factor here for both the cones be $r_1$ and $r_2$ .

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 cones from the above theorem is:

  $\begin{array}{c}\text{Volumes}={\left(\text{radius}\right)}^3={\left(\text{scale}\text{factor}\right)}^3\\ \frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^3\end{array}$

Substitute $8\pi$ for $r_1$ and $27\pi$ for $r_2$ in the above expression.

  $\frac{8\pi }{27\pi }={\left(\frac{r_1}{r_2}\right)}^3$

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

  $\begin{array}{c}\sqrt[3]{\frac{8\pi }{27\pi }}=\sqrt[3]{{\left(\frac{r_1}{r_2}\right)}^3}\\ \frac{r_1}{r_2}=\frac{2}{3}\end{array}$

Therefore, the ratios of radii of two similar cones is $2:3$ .

(b).

To determine

To find:The ratios of slant heights of two similar cones.

Answer to Problem 7WE

The ratios of slant heights of two similar conesis $4:9$ .

Explanation of Solution

Given information:

The volumes of two similar cones are $8\pi$ and $27\pi$ .

Calculation:

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 $a^2:b^2$ .

Let the two Volumes of cones be $V_1$ and $V_2$ and scale factor here is slant height for both the conesbe $l_1$ and $l_2$ .

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 cones from the above theorem is:

  $\begin{array}{c}\text{Volumes}={\left(\text{slant}\text{height}\right)}^3={\left(\text{scale}\text{factor}\right)}^3\\ \frac{V_1}{V_2}={\left(\frac{l_1}{l_2}\right)}^3\end{array}$

Substitute $8\pi$ for $l_1$ and $27\pi$ for $l_2$ in the above expression.

  $\frac{8\pi }{27\pi }={\left(\frac{l_1}{l_2}\right)}^3$

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

  $\begin{array}{c}\sqrt[3]{\frac{8\pi }{27\pi }}=\sqrt[3]{{\left(\frac{l_1}{l_2}\right)}^3}\\ \frac{l_1}{l_2}=\frac{2}{3}\end{array}$

Therefore, the ratios of the slant heights of two similar cones is $2:3$ .

(c).

To determine

To find:The ratios of the lateral areas of two similar cones.

Answer to Problem 7WE

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

Explanation of Solution

Given information:

The volumes of two similar cones are $8\pi$ and $27\pi$ .

Calculation:

As calculated in part(a), The value of $r_1$ is $2$ and $r_2$ 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 lateral area of two similar cones from the above theorem is:

  $\begin{array}{c}\text{lateral}\text{area}={\left(\text{ratio}\text{of}radii\right)}^2={\left(\text{scale}\text{factor}\right)}^2\\ \frac{l.a_1}{l.a_2}={\left(\frac{r_1}{r_2}\right)}^2\end{array}$

Substitute $2$ for $r_1$ and $3$ for $r_2$ 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 the lateral areas of two similar cones is $4:9$ .