Chapter 12.5, Problem 6WE

(a)

To determine

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

Answer to Problem 6WE

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

Explanation of Solution

Given information:

The lateral areas of two similar cylinders are $81\pi$ and $144\pi$ .

Calculation:

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

Let the two lateral areas of cylinders be $l.a_1$ and $l.a_2$ and scale factor here for both the cylinders 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 base areas, lateral areas and total areas is $a^2:b^2$ .

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

  $\begin{array}{c}\text{lateral}\text{area}={\left(\text{ratio}\text{of}\text{heights}\right)}^2={\left(\text{scale}\text{factor}\right)}^2\\ \frac{l.a_1}{l.a_2}={\left(\frac{h_1}{h_2}\right)}^2\end{array}$

Substitute $81\pi$ for $l.a_1$ and $144\pi$ for $l.a_2$ in the above expression.

  $\frac{81\pi }{144\pi }={\left(\frac{h_1}{h_2}\right)}^2$

To determine the ratio of heights, apply square rooting at both sides in the above expression.

  $\begin{array}{c}\sqrt{\frac{81\pi }{144\pi }}=\sqrt{{\left(\frac{h_1}{h_2}\right)}^2}\\ \frac{h_1}{h_2}=\frac{9}{12}\end{array}$

Divide the numerator and denominator at the right hand side of the above expression by $3$ .

  $\frac{h_1}{h_2}=\frac{9}{12}\frac{÷}{÷}\frac{3}{3}=\frac{3}{4}$

Therefore, the ratios of the heights of two similar cylindersis $3:4$ .

(b)

To determine

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

Answer to Problem 6WE

The ratios of total areas of two similar cylindersis $9:16$ .

Explanation of Solution

Given information:

The lateral areas of two similar cylinders are $81\pi$ and $144\pi$ .

Calculation:

As calculated in part(a), The value of $h_1$ is $3$ and $h_2$ is $4$ .

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 cylinders 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 $3$ for $h_1$ and $4$ for $h_2$ in the above expression.

  $\begin{array}{c}\frac{t.a_1}{t.a_2}={\left(\frac{3}{4}\right)}^2\\ =\frac{3\times 3}{4\times 4}\\ =\frac{9}{16}\end{array}$

Therefore, the ratios of total areas of two similar cylindersis $9:16$ .

(c)

To determine

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

Answer to Problem 6WE

The ratios of volumes of two similar cylindersis $27:64$ .

Explanation of Solution

Given information:

The lateral areas of two similar cylinders are $81\pi$ and $144\pi$ .

Calculation:

As calculated in part(a), The value of $h_1$ is $3$ and $h_2$ is $4$ .

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 cylinders from the above theorem is:

  $\begin{array}{c}\text{Volumes}={\left(\text{scale}\text{factor}\right)}^3\\ \frac{V_1}{V_2}={\left(\frac{h_1}{h_2}\right)}^3\end{array}$

Substitute $3$ for $h_1$ and $4$ for $h_2$ in the above expression.

  $\begin{array}{c}\frac{V_1}{V_2}={\left(\frac{3}{4}\right)}^3\\ =\frac{3\times 3\times 3}{4\times 4\times 4}\\ =\frac{27}{64}\end{array}$

Therefore, the ratios of volumes of two similar cylindersis $27:64$ .