# water is flowing from a shallow concrete conical reservoir...

• Sep 20th 2009, 10:24 AM
genlovesmusic09
water is flowing from a shallow concrete conical reservoir...
Water is flowing at a rate of 50 m^3/min from a shallow concrete conical reservoir, vertex down, of base radius 40m and height 6m. How fast in cm per minutes, is the water level falling when the water is 3ft deep?

I know the volume of a cone is V = (pi/3)r²h
and I know to use the chain rule: dV/dt = (dh/dt)(dV/dh)

but I don't know how to find dh/dt or dv/dh
• Sep 20th 2009, 11:23 AM
skeeter
Quote:

Originally Posted by genlovesmusic09
Water is flowing at a rate of 50 m^3/min from a shallow concrete conical reservoir, vertex down, of base radius 40m and height 6m. How fast in cm per minutes, is the water level falling when the water is 3ft deep?

I know the volume of a cone is V = (pi/3)r²h
and I know to use the chain rule: dV/dt = (dh/dt)(dV/dh)

but I don't know how to find dh/dt or dv/dh

$\displaystyle \frac{r}{h} = \frac{40}{6} = \frac{20}{3}$

$\displaystyle r = \frac{20h}{3}$

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

$\displaystyle V = \frac{\pi}{3} \left(\frac{20h}{3}\right)^2 h$

$\displaystyle V = \frac{400\pi}{27} h^3$

take the time derivative of the last equation to get the relationship between $\displaystyle \frac{dV}{dt}$ and what you are looking for ... $\displaystyle \frac{dh}{dt}$
• Sep 20th 2009, 02:35 PM
genlovesmusic09
$\displaystyle \frac{{dv}} {{dt}} = \frac{{400\pi }} {9}{h^2}\frac{{dh}} {{dt}}$

i know you plug 50 into dv/dt but I'm not sure what to plug in for h... 6m or 3ft?

and i know after that you solve for dh/dt
• Sep 20th 2009, 02:47 PM
skeeter
Quote:

Originally Posted by genlovesmusic09
$\displaystyle \frac{{dv}} {{dt}} = \frac{{400\pi }} {9}{h^2}\frac{{dh}} {{dt}}$

i know you plug 50 into dv/dt but I'm not sure what to plug in for h... 6m or 3ft?

and i know after that you solve for dh/dt

the question asks for dh/dt in cm/min when h = 3 ft(?) ... (sure it's in feet? if so, then you'll have to convert to meters to get it in units of m/min)

I wouldn't convert m to cm until the very end.
• Sep 20th 2009, 02:50 PM
genlovesmusic09
yes the question says "How fast in cm per minutes, is the water level falling when the water is 3ft deep?"
so I don't need to incorporate the 3ft into the equation?
• Sep 20th 2009, 02:53 PM
skeeter
Quote:

Originally Posted by genlovesmusic09
yes the question says "How fast in cm per minutes, is the water level falling when the water is 3ft deep?"
so I don't need to incorporate the 3ft into the equation?

yes you do ... h = 3ft in the derivative equation. you'll need to convert to meters like I told you previously.
• Sep 20th 2009, 02:53 PM
genlovesmusic09
so i plugged 3ft or 0.9144meters into h^2
and found$\displaystyle \frac{{dh}} {{dt}} = \frac{{450}} {{365.76\pi }}$