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. $$\begin{array}{|c|c|c|c|c|c|}\hline \text { Time (h) } & {0} & {1} & {2} & {3} & {4} \\ \hline \text { Leakage (gal/h) } & {50} & {70} & {97} & {136} & {190} \\ \hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline \text { Time (h) } & {5} & {6} & {7} & {8} \\ \hline \text { Leakage (gallh) } & {265} & {369} & {516} & {720} \\ \hline\end{array}$$ 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 gal $/ \mathrm{h}$ after the first 8 hours. If the tanker originally contained $25,000$ gal of oil, approxi- mately 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.
$$\begin{array}{|c|c|c|c|c|c|}\hline \text { Time (h) } & {0} & {1} & {2} & {3} & {4} \\ \hline \text { Leakage (gal/h) } & {50} & {70} & {97} & {136} & {190} \\ \hline\end{array}$$
$$\begin{array}{|c|c|c|c|c|}\hline \text { Time (h) } & {5} & {6} & {7} & {8} \\ \hline \text { Leakage (gallh) } & {265} & {369} & {516} & {720} \\ \hline\end{array}$$
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 gal $/ \mathrm{h}$ after the first 8 hours.
If the tanker originally contained $25,000$ gal of oil, approxi-
mately how many more hours will elapse in the worst case
before all the oil has spilled? In the best case?

Key Concepts

Rate of Change

This concept refers to the measure of how a quantity changes over time. In the context of spillage, the rate of oil leakage is observed and recorded at various time intervals. Understanding the rate of change is essential for determining how quickly the total amount of spilled oil accumulates, which is fundamental in both estimating immediate impacts and predicting future outcomes.

Numerical Integration (Riemann Sums)

Numerical integration is a method for estimating the total accumulation of a quantity that changes continuously over time by summing approximated contributions over subintervals. In this context, the leakage rates at specific times are used to approximate the total oil spilt by summing the contributions over each time interval, thereby approximating the area under the leakage rate curve.

Upper and Lower Riemann Sums

The concepts of upper and lower sums are used to provide estimates for the integral (total spilt oil) when dealing with discrete data points. The lower sum estimates the total by taking the minimum leakage rate on an interval, while the upper sum uses the maximum rate. These methods provide bounds for the actual total quantity spilled, reflecting worst-case and best-case scenarios based on the data.

Extrapolation and Constant Rate Assumption

Extrapolation involves using known data to predict future behavior beyond the measured interval. In cases where the leakage rate becomes constant after a certain time, this assumption allows for further estimates regarding the additional time required for a complete spill from a finite reservoir. By combining the estimated accumulated spill with the constant leakage rate, one can solve for the remaining time until the entire quantity is spilled.

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