Question

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?

Short answer

The total quantity of leaked oil after 5 hours is approximately between 443 and 658 gallons, and after 8 hours, it is approximately between 2271 and 3119 gallons. The tanker will have completely leaked all its oil after 30.4 to 160.9 hours (or about 1.3 to 6.7 days)

Step-by-step solution

Lower and upper estimate for 5 hours

The lower estimate after 5 hours is approximated by the sum of the leakages at the beginning of each hour, while the upper estimate is approximated by the sum of leakages at the end of each hour. Thus, the total quantity of leaked oil will be between \(\sum_{i=0}^{4} L_{i}\) and \(\sum_{i=1}^{5} L_{i}\), where \(L_{i}\) refers to the leakage at hour \(i\). Calculating these sums gives 443 and 658 gallons.

Lower and upper estimate for 8 hours

This process is repeated with the total quantity of leaked oil being between \(\sum_{i=0}^{7} L_{i}\) and \(\sum_{i=1}^{8} L_{i}\). Calculating gives 2271 and 3119 gallons.

Calculate remaining oil after 8 hours

Subtract the worst-case leakage after 8 hours from the original amount of oil in the tanker (25000 gal), giving \(25000 - 3119 = 21881\) gallons remaining.

Calculate time for all oil to leak out

Knowing that the tanker is leaking at a rate of 720 gal/h, calculate the time for all remaining oil to leak out in the best and worst cases, respectively, by dividing the total oil remaining by the leak rate for the best and worst cases. This gives \(21881/136 = 160.9\) hours and \(21881/720 = 30.4\) hours.

Key concepts

Summation Notation

Summation notation is a mathematical shorthand used to express the sum of a sequence of numbers or terms. At its core, it provides a concise way to add up a series of elements in a straightforward manner. For example, consider that you have a sequence of numbers representing the amount of oil leaking from a tanker at each hour. Without summation notation, listing each value and adding them up would be cumbersome, especially as the list grows longer.

In calculus and in our tanker problem, we express this as \( \sum_{i=a}^{b} L_{i} \) where \( L_{i} \) represents the ith term in our leakage problem, 'a' is the starting index, and 'b' is the ending index. This powerful notation allows students to efficiently calculate total leakages – both lower and upper estimates – by summing specific terms without manually adding each term. To illustrate, the lower estimate for the total quantity of oil after 5 hours is the sum of leakages at the beginning of each hour, represented as \( \sum_{i=0}^{4} L_{i} \) and calculated as 443 gallons.

Rate of Change

The rate of change is a concept that reveals how a quantity changes with respect to another variable. In calculus, this is often represented as a derivative. However, in simpler scenarios like the oil spill from a tanker, we can think of the rate of change as the leakage rate per hour. This rate can vary, increasing or decreasing, which is crucial for determining the amount of oil leaked over time.

Analyzing the provided data shows that the rate is accelerating, which is a critical insight for projecting future leakage and managing the crisis effectively. By recognizing that the future leak rate is a constant 720 gallons per hour - after the initial 8-hour period (from Step 3) - we are well-equipped to calculate the best and worst-case scenarios for how much longer the tanker will leak before it empties, which aids in response planning.

Optimization Problems

Optimization problems are about finding the most efficient solution under given constraints. In calculus, this usually means locating the maximum or minimum value of a function. For real-world situations like the leaking oil tanker, optimization can be applied to minimize environmental impact or to estimate the timeframe for clean-up operations effectively.

By calculating lower and upper estimates for the oil leakage over time, for both 5 hours and 8 hours, we're using principles of optimization to predict outcomes within a certain range. The aim is to prepare for the worst-case scenario while hoping for the best outcome. Step 4 of the solution uses optimization concepts to estimate, given the rate of oil leakage, how many more hours will elapse before the tanker is empty, allowing responders to optimize their strategies accordingly.