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

**genlovesmusic09** · September 20th 2009, 10:24 AM

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

**skeeter** · September 20th 2009, 11:23 AM

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

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

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

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

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

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

take the time derivative of the last equation to get the relationship between $\frac{dV}{dt}$ and what you are looking for ... $\frac{dh}{dt}$

**genlovesmusic09** · September 20th 2009, 02:35 PM

$\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

**skeeter** · September 20th 2009, 02:47 PM

Originally Posted by genlovesmusic09

$\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.

**genlovesmusic09** · September 20th 2009, 02:50 PM

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?

**skeeter** · September 20th 2009, 02:53 PM

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.

**genlovesmusic09** · September 20th 2009, 02:53 PM

so i plugged 3ft or 0.9144meters into h^2
and found $\frac{{dh}} {{dt}} = \frac{{450}} {{365.76\pi }} $

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