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# A cone a hemisphere and a cylinder stand on equal bases and have the same height. Find the ratio of their volumes

Manan Jain
12 Points
4 years ago
We know that all the three figures-cone,hemisphere and cylinder are on same base. It means that they will have the same base radius.
so, let the common radius of all figures be ‘r’.
also they have same height.
so, let the common heights of the three figures be ‘h’.

Now, volume of hemisphere=$\inline \frac{2}{3}\pi r^{3}$
volume of cylinder=$\inline \pi$r2h
volume of cone=$\inline \frac{1}{3}\pi r^{2}h$

so, volume of cone:volume of cylinder:volume of hemisphere=$\inline \frac{2}{3}\pi r^{3}$:$\inline \pi$r2h:$\inline \frac{1}{3}\pi r^{2}h$

{{(Cancel out the common values)}}
=$\inline \frac{2}{3}$r$\inline \pi$×r×r:$\inline \pi$r×r×h:$\inline \frac{1}{3}$$\inline \pi$r×r×h
=$\inline \frac{2}{3}$r:h:$\inline \frac{1}{3}$h
we know that in a hemisphere, radius=height of hemisphere,
so,
=$\inline \frac{2}{3}$h:h:$\inline \frac{1}{3}$h
=$\inline \frac{2}{3}$:1:$\inline \frac{1}{3}$
multiply the ratio by 3,we get
=2:3:1