Chapter 12.3, Problem 29WE

a.

To determine

To determine the volume and lateral area of each prism

Answer to Problem 29WE

The volume and lateral area of the prism is $270\sqrt{3}$ cu.cm and $180\sqrt{3}$ sq.cm

Explanation of Solution

Given: Consider thegiven information,

Radius of cylinder = 6cm

Height of cylinder = 10cm

Formula Used:

Volume of the prism = $a^2\frac{h}{3}$

Lateral area of the prism = $ph$ ( p = perimeter)

Calculation: Let the edge of the base = a

Then, $\cos {30}^{\circ }=\frac{a/2}{R}$ ( $R$ is radius)

  $\frac{\sqrt{3}}{2}=\frac{a}{2R}$

  $a=R\sqrt{3}$

Volume = $\frac{\sqrt{3}}{4}\times (a^2)\times 10$

  = $\frac{\sqrt{3}}{4}\times (3R^2)\times 10$

  = $\frac{5\times 3\sqrt{3}}{2}{(6)}^2$

  = $\frac{15\sqrt{3}}{2}\times 36$

  = $270\sqrt{3}$ cu.cm

Therefore, the volume of prism= $270\sqrt{3}$

Lateral area $=3\times 6\sqrt{3}\times 10$

  $=180\sqrt{3}$ sq.cm

Conclusion:

The volume and lateral area of the prism is $270\sqrt{3}$ cu.cm and $180\sqrt{3}$ sq.cm

b.

To determine

To determine the volume and lateral area of the prism

Answer to Problem 29WE

The volume and lateral area of the prism is $720$ cu.cm and $180\sqrt{2}$ sq.cm

Explanation of Solution

Given: Consider thegiven information,

Radius of cylinder = 6cm

Height of cylinder = 10cm

Formula Used:

Volume of the prism = $a^2\frac{h}{3}$

Lateral area of the prism = $ph$ ( p = perimeter)

Calculation: Let the edge of the base = a

Then, $a^2=R^2+R^2=2R^2$

  $a=\sqrt{2}R$

Volume $=a^2\times 10=20R^2=720$ cu.cm

Therefore, the volume of prism = 720 cu.cm

Lateral area $=3\times \sqrt{2}\times R\times 10$

  $=180\sqrt{2}$ sq.cm

Conclusion:

The volume and lateral area of the prism is $720$ cu.cm and $180\sqrt{2}$ sq.cm

c.

To determine

To determine the volume and lateral area of each prism

Answer to Problem 29WE

The volume and lateral area of the prism is $90\sqrt{3}$ cu.cm and $180$ sq.cm

Explanation of Solution

Given: Consider thegiven information,

Radius of cylinder = 6cm

Height of cylinder = 10cm

Formula Used:

Volume of the prism = $a^2\frac{h}{3}$

Lateral area of the prism = $ph$ ( p = perimeter)

Calculation: Let the edge of the base = a

Then, $\cos {60}^{\circ }=\frac{a/2}{R}$ ( $R$ is radius)

  $\frac{1}{2}=\frac{a}{2R}$

  $a=R$

Volume = $\frac{\sqrt{3}}{4}\times (a^2)\times 10$

  = $\frac{\sqrt{3}}{4}\times (R^2)\times 10$

  = $\frac{5\times \sqrt{3}}{2}\times {(6)}^2$

  = $\frac{5\sqrt{3}}{2}\times 36$

  = $90\sqrt{3}$ cu.cm

Therefore, the volume of prism = $90\sqrt{3}$ cu.cm

Lateral area $=3\times 6\times 10=180$ sq.cm

Conclusion:

The volume and lateral area of the prism is $90\sqrt{3}$ cu.cm and $180$ sq.cm