Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed, and values of the rate at two hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.V ≈ L (lower estimate)V ≈ L (upper estimate)t (h) 0 2 4 6 8 10r(t) (L/h) 8.6 7.6 6.7 6.4 5.6 5.3
Question
Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed, and values of the rate at two hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.V ≈ L (lower estimate)V ≈ L (upper estimate)t (h) 0 2 4 6 8 10r(t) (L/h) 8.6 7.6 6.7 6.4 5.6 5.3
Solution
The problem is asking for lower and upper estimates for the total amount of oil that leaked out. This can be done by calculating the area of the rectangles under the curve of the rate function r(t).
The lower estimate is calculated by taking the minimum height of each interval (which is the rate at the end of each interval) and multiplying it by the width of the interval (which is 2 hours).
The upper estimate is calculated by taking the maximum height of each interval (which is the rate at the beginning of each interval) and multiplying it by the width of the interval (which is 2 hours).
Lower estimate: V ≈ (2 hours)(7.6 L/h) + (2 hours)(6.7 L/h) + (2 hours)(6.4 L/h) + (2 hours)(5.6 L/h) + (2 hours)(5.3 L/h) = 63.2 L
Upper estimate: V ≈ (2 hours)(8.6 L/h) + (2 hours)(7.6 L/h) + (2 hours)(6.7 L/h) + (2 hours)(6.4 L/h) + (2 hours)(5.6 L/h) = 69.8 L
So, the lower estimate for the total amount of oil that leaked out is 63.2 liters and the upper estimate is 69.8 liters.
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