Problem 19
Question
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. $$\begin{array}{l|l|l|l|l|l|} \text { Time (h) } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Leakage (gal /h) } & 50 & 70 & 97 & 136 & 190 \end{array}$$ $$\begin{array}{l|c|c|c|c|} \text { Time (h) } & 5 & 6 & 7 & 8 \\ \hline \text { Leakage (gal/h) } & 265 & 369 & 516 & 720 \end{array}$$ 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 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?
Step-by-Step Solution
Verified Answer
After 5 hours: 543 to 758 gallons. After 8 hours: 1693 to 2363 gallons. More time till full leak: Worst case 31 more hours, best case 33 more hours.
1Step 1: Understanding the Problem
We need to find estimates for the total oil leakage in certain time frames and predict how long it will take for the tanker to fully discharge its oil based on worsening damage rates over time.
2Step 2: Estimating Leakage After 5 Hours - Lower Estimate
For the lower estimate, assume the leakage rate remains constant within each one-hour interval at the start of the interval. Calculate the total leakage as follows: \( 50(1) + 70(1) + 97(1) + 136(1) + 190(1) = 543 \) gallons.
3Step 3: Estimating Leakage After 5 Hours - Upper Estimate
For the upper estimate, assume the leakage rate remains constant at the end of each one-hour interval. Calculate as follows: \( 70(1) + 97(1) + 136(1) + 190(1) + 265(1) = 758 \) gallons.
4Step 4: Estimating Leakage After 8 Hours - Lower Estimate
Using the same procedure as before: \( 50(1) + 70(1) + 97(1) + 136(1) + 190(1) + 265(1) + 369(1) + 516(1) = 1693 \) gallons.
5Step 5: Estimating Leakage After 8 Hours - Upper Estimate
For upper estimate till 8 hours: \( 70(1) + 97(1) + 136(1) + 190(1) + 265(1) + 369(1) + 516(1) + 720(1) = 2363 \) gallons.
6Step 6: Calculating Remaining Time until Full Leak - Worse Case
In the worst case, the tanker continues to leak at the maximal rate of 720 gal/h. Calculate time needed: \( \frac{25000 - 2363}{720} \approx 31 \) more hours.
7Step 7: Calculating Remaining Time until Full Leak - Best Case
In the best case, assume the leakage slows after the initial 8-hour burst. Use the end of 8-hour estimate of oil leaked as a base. Calculate: \( \frac{25000 - 1693}{720} \approx 33 \) more hours.
Key Concepts
Upper and Lower EstimatesOil Leakage ProblemRate of Change
Upper and Lower Estimates
When dealing with dynamic situations like the leakage of oil over time, we often need to make predictions or estimates.
To do this effectively, we use upper and lower estimates which provide us with a range of possibilities.
These estimates help us understand the possible extents of a scenario, which can guide decisions and planning. In this exercise: - **Lower Estimate**: Assume the oil leakage rate remains steady at the beginning of each time interval.
This gives us a more conservative estimation of how much oil leaks out.
For instance, after 5 hours, we calculate by summing the products of each initial leakage rate with the time interval: \( 50(1) + 70(1) + 97(1) + 136(1) + 190(1) = 543 \) gallons.
- **Upper Estimate**: Assume the rate holds constant at the end of each interval, providing a more generous leakage assessment.
This accounts for the maximum possible leakage: \( 70(1) + 97(1) + 136(1) + 190(1) + 265(1) = 758 \) gallons after 5 hours.
These estimates give us bounds within which the actual leakage lies, crucial in planning and mitigating the effects of oil spillages.
To do this effectively, we use upper and lower estimates which provide us with a range of possibilities.
These estimates help us understand the possible extents of a scenario, which can guide decisions and planning. In this exercise: - **Lower Estimate**: Assume the oil leakage rate remains steady at the beginning of each time interval.
This gives us a more conservative estimation of how much oil leaks out.
For instance, after 5 hours, we calculate by summing the products of each initial leakage rate with the time interval: \( 50(1) + 70(1) + 97(1) + 136(1) + 190(1) = 543 \) gallons.
- **Upper Estimate**: Assume the rate holds constant at the end of each interval, providing a more generous leakage assessment.
This accounts for the maximum possible leakage: \( 70(1) + 97(1) + 136(1) + 190(1) + 265(1) = 758 \) gallons after 5 hours.
These estimates give us bounds within which the actual leakage lies, crucial in planning and mitigating the effects of oil spillages.
Oil Leakage Problem
Oil leakages are environmental emergencies that require immediate and thoughtful response.
They involve dynamic conditions where leakage rates can fluctuate due to factors like worsening damages to the tanker.
In the problem at hand, the leakage rate increases noticeably over time: - Starting at 50 gallons per hour at 0 hours, and reaching up to 720 gallons per hour by 8 hours.
The problem involves creating reasonable estimates for the amount of oil escaping over different time frames, which is a practical approach to managing environmental damage. Through this approach: - We focus not only on the current leakage but predict the situation as it might develop, crucial for timely intervention and resource allocation.
By considering both the worst and best-case scenarios, responders can better prepare for varying circumstances that can arise.
They involve dynamic conditions where leakage rates can fluctuate due to factors like worsening damages to the tanker.
In the problem at hand, the leakage rate increases noticeably over time: - Starting at 50 gallons per hour at 0 hours, and reaching up to 720 gallons per hour by 8 hours.
The problem involves creating reasonable estimates for the amount of oil escaping over different time frames, which is a practical approach to managing environmental damage. Through this approach: - We focus not only on the current leakage but predict the situation as it might develop, crucial for timely intervention and resource allocation.
By considering both the worst and best-case scenarios, responders can better prepare for varying circumstances that can arise.
Rate of Change
The concept of rate of change is central to understanding how quickly something alters over time.
In this context, it describes how fast the oil is leaking from the tanker, with rates provided for specific times. This rate increases as: - From 50 gallons per hour initially, up to 720 gallons per hour after 8 hours, indicating a rapid acceleration in leakage. Calculating rates of change helps in: - Predicting future states and planning accordingly.
For the oil tanker, knowing the rate at which leakage is increasing allows estimations for timeframes in which it would entirely empty, both in worst-case and best-case scenarios. For example, in the worst-case scenario: - After calculating the oil discharged by the 8-hour mark, it assumes the leakage continues at 720 gallons per hour.
Therefore, approximately 31 more hours are needed for complete spillage from a starting 25,000 gallons of oil. This analysis aids in devising response strategies to manage or mitigate the consequences effectively.
In this context, it describes how fast the oil is leaking from the tanker, with rates provided for specific times. This rate increases as: - From 50 gallons per hour initially, up to 720 gallons per hour after 8 hours, indicating a rapid acceleration in leakage. Calculating rates of change helps in: - Predicting future states and planning accordingly.
For the oil tanker, knowing the rate at which leakage is increasing allows estimations for timeframes in which it would entirely empty, both in worst-case and best-case scenarios. For example, in the worst-case scenario: - After calculating the oil discharged by the 8-hour mark, it assumes the leakage continues at 720 gallons per hour.
Therefore, approximately 31 more hours are needed for complete spillage from a starting 25,000 gallons of oil. This analysis aids in devising response strategies to manage or mitigate the consequences effectively.
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