Chapter 12.5, Problem 34PPS

To determine

To find the volume of the given cone.

Answer to Problem 34PPS

  $V=70$ cubic millimeters (approximately)

Explanation of Solution

Given:

  

Slant height ( $l$ ) $=6$ millimeters, Lateral surface area $=71.6$ square millimeters

Formula used:

Lateral surface area of cone $=\pi rl$

where $l$ is the slant height of the cone.

Volume of cone, $V=\frac{1}{3}Bh=\frac{1}{3}\pi r^{2}h$

where $B=\pi r^2$ is the area of the base and $h$ is the height of the cone.

Pythagoras Theorem:

In right angle $\Delta ABC$

  $A{B}^2+B{C}^2=A{C}^2$

Calculation:

Lateral surface area of the given cone

  $\Rightarrow 71.6=\pi rl$

  $\Rightarrow r=\frac{71.6}{\pi l}$

  $\Rightarrow r=\frac{71.6}{3.14\times 6}$

  $\Rightarrow r=3.8$ millimeters

  

In right angled $\Delta ABC$,$A{B}^2+B{C}^2=A{C}^2$

Now putting the values,

  $\Rightarrow A{B}^2+{3.8}^2=6^2$

  $\Rightarrow A{B}^2=6^2-{3.8}^2$

  $\Rightarrow A{B}^2=36-14.44$

  $\Rightarrow A{B}^2=21.56$

  $\Rightarrow AB=\sqrt{21.56}$

  $\Rightarrow AB=h=4.6$ millimeters

For the volume ( $V$ ) of the cone,

  $V=\frac{1}{3}Bh=\frac{1}{3}\pi r^{2}h$

Put the values of $r=3.8$ and $h=4.6$ ,

  $\Rightarrow \frac{1}{3}\times 3.14\times {(3.8)}^2\times 4.6$$\left[\pi =3.14\right]$

  $\Rightarrow \frac{208.6}{3}$

  $\Rightarrow V=69.52$ cubic millimeters

Conclusion:

Therefore, the volume of the given cone is $70$ cubic millimeters (approximately).