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) 8.7 7.9 6.7 6.2 5.7 5.1
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) 8.7 7.9 6.7 6.2 5.7 5.1
Solution
The given table represents the rate of oil leakage at different time intervals. To find the lower and upper estimates for the total amount of oil leaked, we can use the concept of Riemann sums in calculus, specifically the Left Hand Rule (LHR) and Right Hand Rule (RHR).
The Left Hand Rule (LHR) uses the left endpoint of each interval to estimate the total. This will give us the lower estimate.
The Right Hand Rule (RHR) uses the right endpoint of each interval to estimate the total. This will give us the upper estimate.
The width of each interval is 2 hours.
Lower estimate (LHR): = (8.7 * 2) + (7.9 * 2) + (6.7 * 2) + (6.2 * 2) + (5.7 * 2) = 34.8 liters
Upper estimate (RHR): = (7.9 * 2) + (6.7 * 2) + (6.2 * 2) + (5.7 * 2) + (5.1 * 2) = 31.6 liters
So, the lower estimate for the total amount of oil that leaked out is approximately 34.8 liters and the upper estimate is approximately 31.6 liters.
Similar Questions
An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r(t) = 110e−0.05t liters per minute. How much oil leaks out during the first hour? (Round your answer to the nearest liter.)
Water flows from the bottom of a storage tank at a rate of r(t) = 300 − 6t liters per minute, where 0 ≤ t ≤ 50. Find the amount of water that flows from the tank during the first 30 minutes.
An industrial cylindrical water tank has a radius of r meters and a height of h meters, where h is twice the radius. When the tank is full, its water level decreases at a rate of 0.25 meters per hour due to usage. After 4 hours of usage, what is the volume of water remaining in the tank in terms of r?
A rectangular water tank measures 2.5m long, 2.4m wide and 2.1m high. The tank contained some water up to a height of 1.21m.An inlet pipe was opened and water let to flow into the tank at a rate of 8 litres per minute. After one hour, a drain pipe was opened and water allowed to flow out of the tank at a rate of 6 litres per minute.Calculatei. The height of water in tank after 3 hours.ii. The total time taken to fill up the tank.
An oil tank has to be drained for maintenance. The tank is shaped like a cylinder that is 4 ft long with a diameter of 2.2 ft. Suppose oil is drained at a rate of 2.3 ft3 per minute. If the tank starts completely full, how many minutes will it take to empty the tank?Use the value 3.14 for π, and round your answer to the nearest minute. Do not round any intermediate computations.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.