Chapter 12.5, Problem 6CYU

To determine

To find the volume of the given cone.

Answer to Problem 6CYU

  $V\approx 15,540$ cubic centimeters (approximately)

Explanation of Solution

Given:

  

  $h=11.5\text{centimeters}$ , $\theta ={18}^0$

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:

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

  $\tan {18}^0=0.32$

Calculation:

  

In $\Delta ABC$ ,

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

  $BC=r$

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

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

  $\Rightarrow \tan {18}^0=\frac{AB}{BC}$

  $\Rightarrow \tan {18}^0=\frac{11.5}{r}$

  $\Rightarrow 0.32=\frac{11.5}{r}$

On cross-multiplying,

  $\Rightarrow r=\frac{11.5}{0.32}$

  $\Rightarrow r=35.93$ centimeters

Now for the volume of the cone,

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

Put the values of $r=35.93$ and $h=11.5$ ,

  $\Rightarrow \frac{1}{3}\times 3.14\times {(35.93)}^2\times 11.5$

  $\left[\pi =3.14\right]$

  $\Rightarrow \frac{46616.7425}{3}$

  $\Rightarrow V\approx 15,540$ cubic centimeters

Conclusion:

Therefore, the volume of the given cone is $15,540$ cubic centimeters (approximately).