Water pollution Oil is leaking out of a tanker damaged at...

Water pollution Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in the following table. a. Give an upper and a lower estimate of the total quantity of oil that has escaped after 5 hours. b. Repeat part (a) for the quantity of oil that has escaped after 8 hours. c. The tanker continues to leak 720 gal $/ \mathrm{h}$ after the first 8 hours. If the tanker originally contained $25,000$ gal of oil, approximately how many more hours will elapse in the worst case before all the oil has spilled? In the best case?

Water pollution Oil is leaking out of a tanker damaged at sea. The
damage to the tanker is worsening as evidenced by the increased
leakage each hour, recorded in the following table.
a. Give an upper and a lower estimate of the total quantity of oil
that has escaped after 5 hours.
b. Repeat part (a) for the quantity of oil that has escaped after
8 hours.
c. The tanker continues to leak 720 gal $/ \mathrm{h}$ after the first 8 hours.
If the tanker originally contained $25,000$ gal of oil, approximately how many more hours will elapse in the worst case
before all the oil has spilled? In the best case?

Key Concepts

Riemann Sums

This concept involves approximating the area under a curve by breaking it into subintervals and summing up areas of simple shapes, often rectangles. It is essential when exact integration is complicated or when only discrete data points are available, providing both upper and lower estimates depending on whether one uses right-endpoints or left-endpoints to determine the rectangle heights.

Definite Integration

Definite integration is the process of finding the total accumulation of a quantity, such as volume or distance, over an interval using the integral of its rate function. It is the continuous analogue of summing many small contributions, which explains how approximated areas (or accumulated quantities) converge to the exact value as the partition is refined.

Rate of Change

This concept refers to the notion of how one quantity changes with respect to another, such as oil spill rate over time. By understanding the instantaneous rate, one can integrate over time to compute the total amount accumulated, highlighting the link between derivative functions and their antiderivatives.

Upper and Lower Estimates

These estimates are strategies used to bound the true value of an integral when using numerical methods. In contexts where data is given discretely, the lower estimate typically involves using the minimum function values on subintervals, while the upper estimate uses the maximum function values, thus creating a range within which the true accumulated quantity lies.

Transcript

00:01 Hello, hello.

00:02 So we're going to be looking at a problem where we're getting an upper and a lower estimate for an oil spill.

00:08 And one of the first things i'm going to make a note of is we've got an increasing function here.

00:16 So the nice thing about an increasing function is that whenever we're choosing our upper, lower number, so if we're choosing for one here, then we would be choosing the right one.

00:30 Because it's increasing function.

00:32 And so you can see as you go on, that's going to continue to be the case.

00:36 And so that's one nice thing about this problem.

00:38 The second nice thing about this problem is that we've got this time and it's going up in increments of one.

00:44 So typically when you're doing an upper lower estimate, you're using this chart to create bars so you can create these estimate areas to calculate your estimates.

00:54 And so with that width of being one, we're basically just adding up what our leakage gallon per hour.

01:00 Numbers are, but really we are multiplying it by that difference of time, but in this case, that difference of time is just one.

01:09 So let's go ahead and start finding our upper estimates, and we'll kind of map it out as well.

01:16 So for this first one here, our upper estimate is going to be 70, because it's the right hand number there.

01:23 So that's our first hour.

01:25 Then our second hour, we're going to be getting 97, third hour, 136.

01:37 Fourth, it's going to be 190, and finally 265.

01:49 And that's gonna end up giving us an answer of 758.

01:55 And so the reason i've got broken off in a and b for upper estimate and a and b for lower estimate is because we can just keep continuing and we'll find out what our upper estimate is for the first eight hours just based on what we found for our first five hours.

02:09 So this first five hours is gonna be exactly the same.

02:13 Still going to have this 758 over here.

02:17 Then we'll be adding up this 369 for that sixth hour, 369, then adding 516 for the seventh, and then we'll be adding 720.

02:35 And so i promise that that up there is a zero.

02:44 Once we add all of these up together, we're going to get 2 ,363.

02:53 And so ta -da, you've got your a and b for the up.

02:56 And now for our lower estimates and so like i mentioned before we have an increasing function and with those increasing functions we are just going to be going with the left number and we've still got that nice width of just one in between or yeah our widths of one as our kind of dx or difference in time or a difference in our x component so we've got this first one's gonna be 50 our second is going to be 70.

03:33 Our third hour is going to be 97.

03:38 Our fourth hour is going to be 136.

03:42 And then our fifth hour is going to be 190.

03:48 And that's going to give us an estimate of $543.

03:58 And so, as stated before, what's really nice about what we're doing is this is our estimate for our first five hours.

04:06 And so all we have to do to get our first eight hours is we have to add the remaining three hours from five to eight.

04:13 So we can start with that 543 as our first five hours.

04:18 And we just add on from this graph that we've been using.

04:22 So 265, 369, and 516.

04:37 And that is going to give us 1 ,693.

04:46 So just kind of a couple notes i want to make is the first one being that this was a very nice problem to work with in terms of finding an upper and lower estimate because of that continuing that increasing property that has.

05:07 So each plot was increasing.

05:08 So that made it really nice...

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