Question

Determine the dimensions of the right circular cylinder of

greatest volume that can be inscribed in a right circular cone of

radius 6 cm and height 9 cm.

Answer 1

Using similar trianglr

Let h and r be the height and radius of cylinder respectively

Volume of cylinde:

$V=\pi r^2h$

$\frac{9}{6}=\frac{h}{6-r}$

$h=\frac{54-9r}{6}$

Put the value of the height into the formula for the volume of cylinder.

$V=\pi \frac{54-9r}{6}r^2=\frac{\pi*54*r^2}{6}-\frac{\pi*9r*r^2}{6}=9\pi*r^2-\frac{\pi*3r^3}{2} V’=18\pi*r-4,5\pi*r^2$

$V’=18\pi*r-4,5\pi*r^2$

$V’=0$

$18\pi*r-4,5\pi*r^2=0$

$9\pi*r(2-0,5r)=0$

$r=0$

and

$r=4$

$h=\frac{54-9*4}{6}=3$

$V=\pi*3*4^2=48\pi(cm^3)$