# "Star Problems" Volume as a function of height

• May 17th 2010, 12:57 PM
wildcurls
"Star Problems" Volume as a function of height
Hello Everyone,

I need help on problems like this one:

A water tank is the shape of an inverted cone which has a height of 120 meters and a diameter at the top of 60 meters. Water starts flowing out of the tank at a rate of 5 cubic meters per second.
Find the volume of the water, V, as a function of water height, H.
Find H as a function of time, T, since the water started flowing from a full tank.

Thanks!
• May 17th 2010, 04:17 PM
skeeter
Quote:

Originally Posted by wildcurls
Hello Everyone,

I need help on problems like this one:

A water tank is the shape of an inverted cone which has a height of 120 meters and a diameter at the top of 60 meters. Water starts flowing out of the tank at a rate of 5 cubic meters per second.
Find the volume of the water, V, as a function of water height, H.
Find H as a function of time, T, since the water started flowing from a full tank.

Thanks!

http://www.mathhelpforum.com/math-he...ted-rates.html
• May 18th 2010, 06:58 AM
wildcurls
Unfortunatly that didn't really help me much. I don't know what derivative of w/r means or the http://www.mathhelpforum.com/math-he...2e23587f-1.gif If some one could explain these it might help me more. My pre-cal teacher is really frustrating, he didn't explain these problems just handed them out. Can someone explain this to me in pre-cal language?
• May 18th 2010, 08:00 AM
sa-ri-ga-ma
Quote:

Originally Posted by wildcurls
Unfortunatly that didn't really help me much. I don't know what derivative of w/r means or the http://www.mathhelpforum.com/math-he...2e23587f-1.gif If some one could explain these it might help me more. My pre-cal teacher is really frustrating, he didn't explain these problems just handed them out. Can someone explain this to me in pre-cal language?

Volume of the cone V = $\displaystyle \frac{1}{3}\pi{r^2h}$

If H is the height of the cone and R is the radius of the base, then $\displaystyle \frac{H}{R} = \frac{h}{r}$

r = $\displaystyle \frac{hR}{H}$

Substitute this value in V to get V in terms of h.
• May 18th 2010, 09:03 AM
skeeter
Quote:

A water tank is the shape of an inverted cone which has a height of 120 meters and a diameter at the top of 60 meters. Water starts flowing out of the tank at a rate of 5 cubic meters per second.
Find the volume of the water, V, as a function of water height, H.
Find H as a function of time, T, since the water started flowing from a full tank.

the water in the tank forms a conical shape similar to the entire cone.

let $\displaystyle r$ = radius of the water surface

$\displaystyle h$ = height of the water surface

$\displaystyle \frac{r}{h} = \frac{30}{120} = \frac{1}{4}$

so, $\displaystyle 4r = h$ or $\displaystyle r = \frac{h}{4}$

let $\displaystyle V$ = volume of water in the tank

$\displaystyle V = \frac{\pi}{3} r^2 h$

substitute $\displaystyle \frac{h}{4}$ for $\displaystyle r$ in the volume equation

$\displaystyle V = \frac{\pi}{3} \left(\frac{h}{4}\right)^2 h$

$\displaystyle V = \frac{\pi}{48} h^3$

since $\displaystyle V = 5t$ ...

$\displaystyle 5t = \frac{\pi}{48} h^3$

to get $\displaystyle h$ as a function of time, solve the last equation for $\displaystyle h$.
• May 19th 2010, 06:09 AM
wildcurls
Thank you!!!