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Problem 52 Hard Difficulty

If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $ V $ of water remaining in the tank after $ t $ minutes as
$$ V(t) = 100,000 (1 - \frac{1}{60}t)^2 \hspace{5mm} 0 \le t \le 60 $$

Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $ V $ with respect to $ t $) as a function of $ t $. What are its units? For times $ t $ = 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? the least?

Answer

flow rate greatest at the start, least at the end

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Video Transcript

okay we have a tank and water is flowing out of it and this is a formula for the volume of it. So wants us to find the flow rate and the units of the flow rate and then plug some numbers in. So let's see what happens. So V. Of T. Means it's the volume with respect to time. So it's related to time. So when we find the flow rate we're taking the derivative with respect to T. So T. Is the variable. Okay then leave mushroom. Er So the prime of T. Or if you want to call it D. V. D. T. The same thing. So the 100,000 is a constant and then you've got the next part is squared. So it's derivative is too times the stuff To the one power times the derivative of the stuff which would be -1/60. Okay, so that gives us um -200000 over 60 Times 1 -1/60 T. And so uh huh. I'm just going to leave it like that because Because okay and VFT remember was 100,000 one minus 1/60 T squared. Okay, so it's zero V. Of zero. When T a zero that's 100,000 and the prime of zero is -200000/60 And v. of 10. that will be 101656. That would be 100,000 times 56 squared. And the prime of 10 That will be 56 -200000 or for 60 times 56. All right, so every time it's gonna be the same Soviet 20 will be 100,000 times 46 And this will be -200 whatever or 60 times four. Sixth. Okay, so let's see what's happening and so forth. Yeah, I can use my phone for calculator. So it might not be great. 200,000 divided by 60. So this one is -333. All right then the next 15 divided by six squared 555, 6 squared Times 100,000. Uh huh. So after 10 minutes now, the volume is already down to 69444. And here we have uh 55 x six times 3333. So the flow rate is about this camera on and off. Hey, Can the next one we have four x 6 squared no Times 100,000. So now it's down to 44444 and then six times 3 333. And then the flow rate is this Mhm. Okay. So when is the flow rate the greatest? Well, it's the greatest that the at the beginning. Okay. And it's negative because it's going out. Okay. And it's the greatest because that's the biggest number. So the most is going out at at the beginning and then as the water gets less than the flow rate gets less. So when will it be the least? Will be the least at 50 or whatever is the last thing we're supposed to plug in? Okay, because it's decreasing what?