Chapter 12.5, Problem 2CCYP

To determine

To find the volume of the given cone.

Answer to Problem 2CCYP

  $V\approx 390$ cubic centimeters (approximately)

Explanation of Solution

Given:

  

  $h=5$ centimeters, $\theta ={30}^0$

Formula used:

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.

Trigonometric formulas:

  $\tan \theta =\frac{perpendicular}{base}$

  $\tan {30}^0=\frac{1}{\sqrt{3}}$

Calculation:

  

In $\Delta ABC$ ,

  $AB=$ base, $BC=$ perpendicular, and $CA=$ hypotenuse.

  $AB=r$

  $\Rightarrow \tan \theta =\frac{perpendicular}{base}$

  $\Rightarrow \tan \theta =\frac{AB}{BC}$

  $\Rightarrow \tan {30}^0=\frac{AB}{BC}$

  $\Rightarrow \tan {30}^0=\frac{5}{r}$

  $\Rightarrow \frac{1}{\sqrt{3}}=\frac{5}{r}$

On cross-multiplying,

  $\Rightarrow r=5\sqrt{3}$

  $\Rightarrow r=8.66$ centi meters $\left[\sqrt{3}=1.73\right]$

Now for the volume ( $V$ ) of the cone,

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

Put the values of $r=8.66$ and $h=5$ ,

  $\Rightarrow \frac{1}{3}\times 3.14\times {(8.66)}^2\times 5$

  $\left[\pi =3.14\right]$

  $\Rightarrow \frac{1177.43092}{3}$

  $\Rightarrow 392.476973$ cubic centimeters

  $\Rightarrow V\approx 390$ cubic centimeters

Conclusion:

Therefore, the volume of the given cone is $390$ cubic centimeters (approximately).