(a) pi * r * l = 1/2 (4 * pi * r^2)
l = 2r
(b) I can't do
(c) I can't do
Thanks
Hello, GAdams!
From the diagram in (a), we can use Pythagorus (as janvdl suggested):(b) Find the perpendicular height, $\displaystyle h$, of the cone in terms of $\displaystyle r$.
. . $\displaystyle h^2 + r^2\:=\:L^2\quad\Rightarrow\quad h \:=\:\sqrt{L^2 - r^2}$
Since $\displaystyle L = 2r$, we have: .$\displaystyle h \:=\:\sqrt{(2r)^2 - r^2} \:=\:\sqrt{3r^2}\quad\Rightarrow\quad\boxed{h \:=\:\sqrt{3}r}$
The volume of a cone is: .$\displaystyle V \:=\:\frac{1}{3}\pi r^2h \:=\:\frac{1}{3}\pi r^2(\sqrt{3}r) \:=\:\frac{\sqrt{3}}{3}\pi r^3$(c) Find the ratio of the volumes of the cone and the hemisphere.
The volume of a hemisphere is: .$\displaystyle \frac{1}{2} \times \frac{4}{3}\pi r^3\:=\:\frac{2}{3}\pi r^3$
The ratio is: .$\displaystyle \frac{V_c}{V_h} \;=\;\frac{\frac{\sqrt{3}}{3}\pi r^3}{\frac{2}{3}\pi r^3} \:=\:\boxed{\frac{\sqrt{3}}{2}}$