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

1. 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

2. 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 ... $\displaystyle \frac{dh}{dt}$

3. $\displaystyle \frac{{dv}} {{dt}} = \frac{{400\pi }} {9}{h^2}\frac{{dh}} {{dt}}$

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

4. 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.

5. 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?

6. 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.

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

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draining concial reservoir

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