Problem 68 Medium Difficulty

Water flows into and out of a storage tank. A graph of the rate of change $ r(t) $ of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time $ t = 0 $ is $ 25,000 L $, use the Midpoint Rule to estimate the amount of water in the tank four days later.

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00:01

Frank L.

00:30

Amrita B.

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Video Transcript

So not knowing exactly what your graph looks drawn one at random here hoping that you'll get the general idea. You're told that you start with 225,000 liters of water and it's slowly flowing into and out of the storage tank. So again I don't really know what your graph it looks like but the idea will be the same. So our F. T. Is your Y. Access your vertical access in your horizontal axis is T. And I believe it's days. Yes days. So I don't know what your intervals are. Your values are down here but you're asked to use the midpoint rule to estimate the amount of water in the tank. So midpoint means if you have these values So for example 1234567. And you need to use the midpoint rule you need to find the halfway between your two points. So that's A plus B divided by two. Whatever these values happen to be. So in this case The midpoint rule between zero and 1 would be one half will change colors, and then 1.5, 2.5 etcetera etcetera. So you need to find the midpoint between the values that you're given. And then using the midpoint rule, that just means you need to make some rectangles based on the Y. Value that goes with that midpoint. So now you're gonna have to figure out What are at .5 is because this will now be the height of your rectangle. So whatever this value turns out to be 0.5 And are 0.5, that's going to be the value that you're going to use in your summation. Now I like to make a table. Sorry, that's a little crooked. So T. And R. Of T. Because I like to have it handy. So again 0.5 1.5 2.53 point five whatever your mid points are and then whatever the Y. Value is and this is what we're going to use in your formula. So again would go up to about where that is and that's gonna be my rectangle. So the height of the rectangle is determined by that midpoint. And then when I plug it in it will give me the height and I'll go through and I'll do that on all of these. Where this will be become the height of my rectangle. So when I estimate The amount of water in the tank four days later. So four days later right here I would do that amount in the tank with equal the length of my rectangle. So I made each of mine one unit one unit one unit one unit or the the width of my rectangle. Or might change in T. So my change in T. And then I need to do all the height. So art 0.5 plus are 1.5 plus Are 2.5 Plus, our 3.5 and then my change in tea. The way I drew it, I don't know what your graph looks like is going to be one. So this will just be one. I just need to add all those values up.