If a cylindrical tank holds 100,000 gallons of water, which...

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?

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?

Transcript

00:01 Okay, we have a tank and water is flowing out of it and this is a formula for the volume of it.

00:09 So it wants us to find the flow rate and the units of the flow rate and then plug some numbers in.

00:18 So let's see what happens.

00:20 So v of t means it's the volume with respect to time.

00:25 So it's related to time.

00:27 So when we find the flow rate, we're taking the derivative with respect to t.

00:30 So t is the variable.

00:33 Didn't leave much room here.

00:35 So v prime of t, or if you want to call it dvdt, same thing.

00:42 So the 100 ,000 is a constant.

00:45 And then you've got the next part is squared, so its derivative is two times the stuff to the one power times the derivative of the stuff, which would be minus 1 over 60.

01:01 Okay, so that gives us minus 200 ,000 over 60 times 1 minus 1 over 60 t.

01:16 And so i'm just going to leave it like that because, because, okay, nvt remember, was 100 ,0001 minus 1 over 60t squared.

01:35 Okay, so at 0, v of 0, when t is 0, that's 100 ,000, and v prime of 0 is minus 200 ,000 over 60, and v of 10, that will be 10, 1 6 ,000, that'll be 100 ,000 times 5 ,6 squared, and v.

02:10 10 ,000 of 10 ,000.

02:12 That will be 5 -6 minus 200 ,000 over 60 times 5 -6.

02:24 All right, so every time it's going to be the same.

02:28 So v of 20 will be 100 ,000 times 4 -6th.

02:36 And this will be minus 200, whatever, or 60 times 4 -6th.

02:43 Okay, so let's see what's happening so forth...

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