Question

Find the volume of the figure: a cone with a square pyramid of the same height cut out. The pyramid has height l, and its square base has area l2.

Answer 1

Answer:

The volume of the figure is $\frac{l^{3}}{3}[\frac{1}{2}\pi-1]\ units^{3}$

Step-by-step explanation:

we know that

The volume of the figure is equal to the volume of the cone minus the volume of the square pyramid

step 1

Find the volume of the cone

The volume of the cone is equal to

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

we have

$r=l\frac{\sqrt{2}}{2}\ units$

$h=l\ units$

substitute

$V=\frac{1}{3}\pi (l\frac{\sqrt{2}}{2})^{2}l$

$V=\frac{1}{3}\pi (\frac{l^{3}}{2} )$

$V=\frac{1}{6}\pi (l^{3})\ units^{3}$

step 2

Find the volume of the square pyramid

$V=\frac{1}{3}Bh$

we have

$B=l^{2}\ units^{2}$

$h=l\ units$

substitute

$V=\frac{1}{3}(l^{2})l$

$V=\frac{1}{3}(l^{3})\ units^{3}$

step 3

Find the difference

$\frac{1}{6}\pi (l^{3})-\frac{1}{3}(l^{3})=\frac{l^{3}}{3}[\frac{1}{2}\pi-1]\ units^{3}$