Chapter 12.3, Problem 17WE

a.

To determine

To determine the ratio of lateral area

Answer to Problem 17WE

The ratio of lateral area is $1:4$

Explanation of Solution

Given: Consider the given information, Ratio of radii is $\frac{4}{8}or\frac{1}{2}$ units

Ratio of height is $\frac{3}{6}or\frac{1}{2}$ units

Formula Used:

Lateral area = $\frac{\pi r_1\sqrt{h_1{}^2+r_1{}^2}}{\pi r_2\sqrt{h_2{}^2+r_2{}^2}}$

Calculation:The lateral area ratio is,

  $\frac{\pi r_1\sqrt{h_1{}^2+r_1{}^2}}{\pi r_2\sqrt{h_2{}^2+r_2{}^2}}$

  = $\frac{1\sqrt{{(4)}^2+{(3)}^2}}{2\sqrt{{(8)}^2+{(6)}^2}}$

  = $\frac{1}{2}\frac{\sqrt{25}}{\sqrt{64+36}}$

  = $\frac{1}{2}\times \frac{5}{10}$

  = $\frac{1}{4}$

Conclusion:

The ratio of lateral area is $1:4$

b.

To determine

To determine the ratio of total area

Answer to Problem 17WE

The ratio of total area is $1:4$

Explanation of Solution

Given:Consider the given information, Ratio of radii is $\frac{4}{8}or\frac{1}{2}$

Ratio of height is $\frac{3}{6}or\frac{1}{2}$

Formula Used:

Total area = $\frac{\pi r_1(r_1+\sqrt{h_1+r_1{}^2})}{\pi r_2(r_2+\sqrt{h_2{}^2+r_2{}^2})}$

Calculation:The lateral area ratio is,

  $\frac{\pi r_1(r_1+\sqrt{h_1+r_1{}^2})}{\pi r_2(r_2+\sqrt{h_2{}^2+r_2{}^2})}$

  = $\frac{1\sqrt{(4)+(5)}}{2\sqrt{(8)+(6)}}$

  = $\frac{1}{2}\times \frac{9}{18}$

  = $\frac{1}{4}$

Conclusion:

The ratio of lateral area is $1:4$

c.

To determine

To determine the ratio of volume

Answer to Problem 17WE

The ratio of volume is $1:8$

Explanation of Solution

Given: Consider the given information, Ratio of radii is $\frac{4}{8}or\frac{1}{2}$

Ratio of height is $\frac{3}{6}or\frac{1}{2}$

Formula Used:

Volume = $\frac{\frac{1}{3}\pi r_1{}^2{h}_1}{\frac{1}{3}\pi r_1{}^2{h}_1}$

Calculation: The lateral area ratio is,

  $\frac{\frac{1}{3}\pi r_1{}^2{h}_1}{\frac{1}{3}\pi r_1{}^2{h}_1}$

  = ${(\frac{1}{2})}^2\times \frac{1}{2}$

  = $\frac{1}{8}$

Conclusion:

The ratio of lateral area is $1:8$