Cone with water in it

**stones44** · October 31st 2007, 05:28 PM

Water is being poured into a conical cup that is 12 cm tall.
a. When the water in the cup is 9 cm deep, what percentage of the cup is filled?
b. When the cup is 75% filled, how deep is the water?

I think it has to do with ratio, since it only gives you the height?
Along with the answer, can someone give me like a cheat sheet way of doing this, with either percentage,/ height,/ radius, etc.

**kalagota** · October 31st 2007, 08:59 PM

Originally Posted by stones44

Water is being poured into a conical cup that is 12 cm tall.
a. When the water in the cup is 9 cm deep, what percentage of the cup is filled?
b. When the cup is 75% filled, how deep is the water?

I think it has to do with ratio, since it only gives you the height?
Along with the answer, can someone give me like a cheat sheet way of doing this, with either percentage,/ height,/ radius, etc.

you may use similar triangles there..
suppose that the radius of the cone is R and the radius of the cone at the hight of 9 ft is r, then the ratio is given by
$\frac{R}{12}=\frac{r}{9}$
which implies that
$r= \frac {9R}{12}$

$V_{cone} = \frac{1}{3}\pi R^2 H$
given th situation, the volume of the cone when full is
$V_{cone, full}=4\pi R^2$

while when if 9 ft full,
$V_{cone,9 ft}=3\pi r^2$

then, the percentage is
$\frac{V_{cone,9 ft}}{V_{cone, full}} = \frac{3\pi r^2}{4\pi R^2}$
$= \frac{3\pi (\frac {9R}{12})^2}{4\pi R^2}$
$= \frac{3\frac{9^2 \pi R^2}{12^2}}{4\pi R^2}$
$= \frac{(3)(9^2)}{(4)(12^2)}$
then multiply by 100%.. Ü

for b)
you are looking for h. so, using similar triangles again, you have
$\frac{r}{h}=\frac{R}{12}$
so that
$r=\frac{hR}{12}$
so, 75% full,
$\frac{V_{cone,h}}{V_{cone,full}}=\frac{3}{4}$
so,
$\frac{\frac{1}{3}h\pi r^2}{4\pi R^2} = \frac{\frac{1}{3}h\pi (\frac{hR}{12})^2}{4\pi R^2} = \frac{3}{4}$
can you continue? you will notice that R^2 will cancel and you can now solve for h. Ü

**stones44** · November 1st 2007, 08:07 AM

dont you have to 2/3 to the volume equation?

im confused how your doing your algebra.....:-/

**kalagota** · November 1st 2007, 10:44 PM

Originally Posted by stones44

dont you have to 2/3 to the volume equation?

im confused how your doing your algebra.....:-/

ahh, yeah.. i was thinking it was a cylinder.. wait, i'll edit it..

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