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 table below. $$\begin{array}{|c|c|c|c|c|}\hline \text { Time (h) } & {0} & {1} & {2} & {3} & {4} \\ \hline \text { Leakage (gal/h) } & {50} & {70} & {97} & {136} & {190}\end{array}$$ $$\begin{array}{|c|cccc}{\text { Time (h) }} & {5} & {6} & {7} & {8} \\ \hline \text { Leakage (gal/h) } & {265} & {369} & {516} & {720}\end{array}$$ (a) Give an upper and 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 of the oil has leaked? 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 table below.
$$\begin{array}{|c|c|c|c|c|}\hline \text { Time (h) } & {0} & {1} & {2} & {3} & {4} \\ \hline \text { Leakage (gal/h) } & {50} & {70} & {97} & {136} & {190}\end{array}$$
$$\begin{array}{|c|cccc}{\text { Time (h) }} & {5} & {6} & {7} & {8} \\ \hline \text { Leakage (gal/h) } & {265} & {369} & {516} & {720}\end{array}$$
(a) Give an upper and 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 of the oil has leaked? in the best case?

Key Concepts

Riemann Sums

Riemann sums involve approximating the integral of a function by dividing the interval into subintervals and summing up rectangles whose heights are based on the function’s values at certain chosen points. This technique allows one to estimate the area under a curve—and in application, the total accumulation of a varying rate—even when only discrete data points are available.

Numerical Integration

Numerical integration is the process of estimating the value of a definite integral using discrete data values rather than an analytic expression. It is particularly useful in practical situations where the exact function is unknown or complicated but can be approximated from measured data, such as estimating total oil leakage over time.

Monotonic Functions and Endpoint Approximations

When a function is known to be monotonic (consistently increasing or decreasing), the choice of endpoints in Riemann sums (left or right) provides a systematic way to get lower and upper bounds for the integral. For an increasing function, using left endpoints yields an underestimate (lower sum), while right endpoints give an overestimate (upper sum).

Linear Extrapolation for Constant Rates

Linear extrapolation involves using a constant rate to estimate future values when conditions stabilize. When a process settles to a constant rate, a linear relationship between time and accumulated quantity can be used to predict outcomes such as the remaining time until a resource is depleted.

Best-case and Worst-case Analysis

Best-case and worst-case analysis establishes bounds on the possible outcomes by considering the extremes of a process. In estimation problems, this approach accounts for uncertainty by providing a range—from the scenario minimizing the impact to that maximizing it—helping in understanding the potential variations in the final result.

Transcript

00:01 So we have some oil that is leaking into the sea and all the oil is leaking at this rate gallons per hour.

00:17 So you want to estimate the quality of the level of approximation at 5 hours.

00:40 And so we want to do the right rectangle approximation also at five hours so as you can see here the the leakage is increasing so that if you're gonna consider the overall increasing so that if you do the left rectangle approximation you're be considering considering an area that goes under the total amount the spoil out because well it's the area under the curve so this one should be under approximation and that one that one is going to go like considering the heights at the right endpoint so it's going to be the green so it's going to be over so we do that sum also we will do per one hour that's the time interval that we have one hour times so this leakage is that is at zero up to four to get the approximate at five right so we have here five we'll be considering the those heights that hide at zero higher at one, high at three, five is actually there, high that four.

02:55 For this last rectangle, it's the height at four that we are looking at.

03:00 So all that being said, it would be 50, plus 70, plus 97, plus 136, plus 190, gallons, per hour so that you get the units that you want, you get gallons and those are 543 gallons.

03:52 For the right rectangle approximation we should take us the initial height the leakage at 1 so we start with 7 plus 92, 7, 7 ,000, 7 ,000 plus 92 plus 136 plus 190 plus 16 plus 165 which is the height at 5 since we're taking the the right heights the heights at their right end points and so we'll be those many gallons and those are 750 p .8 gallons.

05:06 Now what we do, now we want to do the left rectangle approximation for the time of 2 8.

05:20 All the same idea applies.

05:23 The seed continues growing.

05:27 So this one will be under...

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