# If a tank holds 5000 gallons of water, which drains from the bttomof the tank in 40 minutes, then torricelli's Law gives the volume Vof water remainin

If a tank holds 5000 gallons of water, which drains from the bttomof the tank in 40 minutes, then torricelli's Law gives the volume Vof water remaining in the tank after t minutes $\left(1-{\frac{t}{40}}^{2}\right)5000=V0\le t\le 40$.
Find the rate at which water is draining from the tank after (a) 5min, (b) 10 min. At what time is the water flowing out the fastest?the slowest?

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Caren

If a tank holds 5000 gallons ofwater, which drains from the bttom of the tank in 40 minutes, thentorricelli's Law gives the volume V of water remaining in the tankafter t minutes
$\left(1-{\frac{t}{40}}^{2}\right)5000=V$
$0\le t\le 40$
IS IT $T⇐\sqrt{40}$?
Find the rate at which water is draining from the tank after (a) 5min, (b) 10 min. At what time is the water flowing out the fastest?the slowest?
WATER FLOWN OUT IN TIME $T=W=5000-5000\left[1-{T}^{2}/40\right]$
DW/DT =$-5000\left(-2T\right)/40=250T$
AT T=5..RATE OF WATER OUT FLOW=DW/DT $=250\ast 5=1250$
AT T=10
$DW/DT=250\ast 10=2500$ GPM
AS PER THE GIVEN EQN.WATER LOW RATE (=250T)IS INCREASING WITHT AND HENCE HIGHEST AT $T=\sqrt{40}$ AND T=0
BUT THIS APPEARS PHYSICALLY FAULTY AND THEGIVEN EQN.APPEARS TO BE DOUBT FULL

Jeffrey Jordon

The rate at which water is draining from the tank is the derivative of V(t). Use the chain rule to find $\frac{dV}{dt}$.

$V\left(t\right)=5000\left(1-\frac{1}{40}t{\right)}^{2}$

Chain rule : $\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)={f}^{\prime }\left(g\left(x\right)\right){g}^{\prime }\left(x\right)$

$f\left(x\right)=5000{x}^{2},g\left(x\right)=1-\frac{x}{40}$

${f}^{\prime }\left(x\right)=2\left(5000\right)x,{g}^{\prime }\left(x\right)=\frac{-1}{40}$

$\frac{dV}{dt}=2\left(5000\right)\left(1-\frac{1}{40}t\right)\left(-\frac{1}{40}\right)=-250\left(1-\frac{1}{40}t\right)$

Step 2

Plug in t=5 to find the flow rate at that point in time. The volume of water in the tank decreases by this amount, so the flow rate must have this magnitude (the negative sign isn't necessary on the answer).