(a) pi * r * l = 1/2 (4 * pi * r^2)

l = 2r

(b) I can't do

(c) I can't do

Thanks

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- Jul 8th 2007, 02:56 AMGAdamsCone and Hemisphere
(a) pi * r * l = 1/2 (4 * pi * r^2)

l = 2r

(b) I can't do

(c) I can't do

Thanks - Jul 8th 2007, 03:10 AMjanvdl
For b)

Use Pyhtagoras.

$\displaystyle h^2 = l^2 + r^2 $

$\displaystyle h = \sqrt{l^2 + r^2} $ - Jul 8th 2007, 08:25 AMSoroban
Hello, GAdams!

Quote:

(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: .$\displaystyle h \:=\:\sqrt{(2r)^2 - r^2} \:=\:\sqrt{3r^2}\quad\Rightarrow\quad\boxed{h \:=\:\sqrt{3}r}$

Quote:

(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: .$\displaystyle \frac{V_c}{V_h} \;=\;\frac{\frac{\sqrt{3}}{3}\pi r^3}{\frac{2}{3}\pi r^3} \:=\:\boxed{\frac{\sqrt{3}}{2}}$

- Jul 8th 2007, 08:44 AMGAdams
The volume of a cone is confusing me:

I can't see how you got from 1/3 * pi * sq.rt3r to sq.rt3/3 * pi * r^3 - Jul 8th 2007, 09:15 AMjanvdl
I tried googling for the formulas in a), but i struggled to find them. So i couldn't prove that formula. I think i struggled to find the hemisphere one.

So if $\displaystyle l = 2r $ then just plug that in like Soroban did.