Chapter 12.5, Problem 16PPS

To determine

To find the height of the given triangular pyramid.

Answer to Problem 16PPS

  $h=18$ centimeters

Explanation of Solution

Given:

  

Volume ( $V$ ) $=144$ cubic centimeters

Right angled triangular base:

Hypotenuse $=10$ centimeters

Base $=8$ centimeters

Formula used:

Volume of pyramid, $V=\frac{1}{3}Bh$

where $B$ is the area of the base and $h$ is the height of the pyramid.

Pythagoras Theorem:

In right angle $\Delta ABC$

  $A{B}^2+B{C}^2=A{C}^2$

Area of a triangle $=\frac{1}{2}\times Base\times Height$

Calculation:

Since the given pyramid is a Right angled triangular pyramid, so it has a right angled triangular base.

  

In right angle $\Delta ABC$ ,

  $AB=$ height, $BC=$ base, and $AC=$ hypotenuse

So, by Pythagoras Theorem,

  $A{B}^2+B{C}^2=A{C}^2$

Now putting the values,

  $\Rightarrow A{B}^2+8^2={(10)}^2$

  $\Rightarrow A{B}^2={(10)}^2-8^2$

  $\Rightarrow A{B}^2=100-64$

  $\Rightarrow A{B}^2=36$

Now, by taking square root on both sides,

  $\Rightarrow AB=6$ centimeters

So, area of the base, $B=\frac{1}{2}\times Base\times Height$

  $\Rightarrow B=\frac{1}{2}\times 8\times 6$

  $\Rightarrow B=24$ square centimeters.

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

  $V=\frac{1}{3}Bh=144$

  $\Rightarrow 144=\frac{1}{3}Bh$

Now putting the values,

  $\Rightarrow 144=\frac{1}{3}\times 24\times h$

  $\Rightarrow 144=8\times h$

  $\Rightarrow h=\frac{144}{8}$

  $\Rightarrow h=18$ centimeters

Conclusion:

Therefore, the height of the given triangular pyramid is $18$ centimeters.