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. (TABLE CAN'T COPY) 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/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.
(TABLE CAN'T COPY)
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/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 In problem number 19, we have an tanker that's leaking oil into the sea, and the damage the tanker is getting worse, so the leakage rate is getting higher and higher as time goes on.

00:13 So part a, we want to give a lower and upper and lower estimate of the total amount of oil that has escaped after five hours.

00:24 So lower estimate, let's do upper estimate first.

00:28 So we'll use the greater value on.

00:31 On each interval, which is going to end up being the same as a right end point sum.

00:37 So we have a, let's do this, gallons of oil.

00:43 It's called gallons.

00:44 This is going to be our delta t, which is our interval width, which is one hour times first five hours.

00:54 So we're going to end up using these values here.

00:57 So we get 70 plus 97 plus 136 plus 190 plus 265.

01:05 And so upper estimate for five hours.

01:09 Run that through the calculator right quick.

01:13 65.

01:14 That's something up.

01:21 758 gallons after five hours.

01:25 Our lower estimate will use these values.

01:30 And we'll say we get 50 plus 70 plus 97 plus 136 plus 190 plus 19.

01:42 So our lower estimate is going to be 543 gallons.

01:51 And part b, we want to do the same thing, but for eight hours instead of five hours.

01:57 So our upper estimate, i'll switch colors here just a minute.

02:01 So for our upper estimate, we'll use these values, and our lower estimate will use these values because that's the lower value on each interval.

02:13 That's confusing to you.

02:14 Just go back and watch some of the previous videos in this chapter, and i explain that in a lot more depth and clarity.

02:22 So here our upper estimate will be our total gallons.

02:29 Oops, let's not do that in red.

02:31 Our total gallons will be our delta t, which is still one.

02:36 And we have 7d plus 97 plus dot dot dot all the way up to 720...

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