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. $$\begin{array}{l|c|c|c|c|c|} \text { Time (h) } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Leakage (L/h) } & 50 & 70 & 97 & 136 & 190 \end{array}$$ $$\begin{array}{l|c|c|c|c|} \text { Time (h) } & 5 & 6 & 7 & 8 \\ \hline \text { Leakage (L/h) } & 265 & 369 & 516 & 720 \end{array}$$ 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 $720 \mathrm{L} / \mathrm{h}$ after the first 8 hours. If the tanker originally contained 25,000 L 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.
$$\begin{array}{l|c|c|c|c|c|}
\text { Time (h) } & 0 & 1 & 2 & 3 & 4 \\
\hline \text { Leakage (L/h) } & 50 & 70 & 97 & 136 & 190
\end{array}$$
$$\begin{array}{l|c|c|c|c|}
\text { Time (h) } & 5 & 6 & 7 & 8 \\
\hline \text { Leakage (L/h) } & 265 & 369 & 516 & 720
\end{array}$$
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 $720 \mathrm{L} / \mathrm{h}$ after the first 8 hours. If the tanker originally contained 25,000 L 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

Scenario Analysis (Worst-case vs Best-case Estimates)

Scenario analysis involves evaluating different possible outcomes based on varying assumptions. In the context of leakage, worst-case estimates assume conditions that maximize leakage (using the higher rate estimates or constant high rates), while best-case estimates use conditions that minimize leakage. This analysis helps in planning and risk management by providing a range of potential outcomes based on the available data.

Definite Integral as Accumulation

The definite integral in calculus represents the total accumulation of a quantity over a given interval. In practical applications, such as calculating the total volume of oil leaked over time, the integral sums the continuously changing rate of leakage to find the overall amount spilled. This concept links the instantaneous rate of change with the total change over time.

Numerical Integration

Numerical integration is the process of estimating the value of a definite integral when an exact form of the function is unavailable or difficult to obtain. It involves approximating the area under a curve by summing the areas of simple shapes, like rectangles, which can be especially useful when dealing with discrete data points, as in measuring cumulative leakage over time.

Riemann Sums

Riemann sums are a fundamental method of numerical integration where the interval of interest is divided into subintervals, and over each subinterval, the function is approximated by a constant value. Using left endpoints provides a lower estimate and right endpoints an upper estimate when the function is increasing. This approach gives a range within which the true value of the integral (or total leaked oil) lies.

Please give Ace some feedback