Water leaks from a tank at the rate of r(t) gallons per hour. The rate decreased as time passed, and values of the rate at two-hour time intervals are shown in the table below. The total amount of water that leaked out is evaluated by a Riemann sum. Find the upper estimate (left end-points of each rectangle) for the total amount of water that leaked out by using five rectangles.
Give your answer with one decimal place.
t(hr) 0- 2- 4- 6- 8- 10
r(t) (gal/hr) 12.0- 10.6- 8.8- 8.0- 6.0- 4.8
area total:
2*12 + 2*10.6+2*8.8+2*8.0+2*6.0
To find the upper estimate for the total amount of water that leaked out using five rectangles, we need to calculate the areas of the rectangles formed under the rate-time curve.
First, let's identify the intervals for each rectangle:
Rectangle 1: Interval [0, 2]
Rectangle 2: Interval [2, 4]
Rectangle 3: Interval [4, 6]
Rectangle 4: Interval [6, 8]
Rectangle 5: Interval [8, 10]
Next, we need to find the heights of the rectangles, which are given by the rate values at the left endpoints:
Rectangle 1: Height = r(0) = 12.0 gal/hr
Rectangle 2: Height = r(2) = 10.6 gal/hr
Rectangle 3: Height = r(4) = 8.8 gal/hr
Rectangle 4: Height = r(6) = 8.0 gal/hr
Rectangle 5: Height = r(8) = 6.0 gal/hr
Now, we can calculate the areas of the rectangles:
Rectangle 1: Area = (2 - 0) * 12.0 = 24.0 gallons
Rectangle 2: Area = (4 - 2) * 10.6 = 21.2 gallons
Rectangle 3: Area = (6 - 4) * 8.8 = 17.6 gallons
Rectangle 4: Area = (8 - 6) * 8.0 = 16.0 gallons
Rectangle 5: Area = (10 - 8) * 6.0 = 12.0 gallons
Finally, we can find the upper estimate (by summing up the areas of the rectangles):
Total upper estimate = Area(Rectangle 1) + Area(Rectangle 2) + Area(Rectangle 3) + Area(Rectangle 4) + Area(Rectangle 5)
Total upper estimate = 24.0 + 21.2 + 17.6 + 16.0 + 12.0 = 90.8 gallons
Therefore, the upper estimate for the total amount of water that leaked out using five rectangles is 90.8 gallons.
To find the upper estimate for the total amount of water that leaked out by using five rectangles, we can use the left end-points of each rectangle.
To calculate the upper estimate, we need to find the maximum value of the rate of leakage in each interval and multiply it by the width of the interval.
First, let's calculate the width of each interval. Since the time intervals are 2 hours each, the width of each interval is 2 hours.
Next, we need to find the maximum value of the rate of leakage in each interval. From the given table, the maximum values for each interval are:
Interval 1: Max rate = 12.0 gal/hr
Interval 2: Max rate = 10.6 gal/hr
Interval 3: Max rate = 8.8 gal/hr
Interval 4: Max rate = 8.0 gal/hr
Interval 5: Max rate = 6.0 gal/hr
Now, we multiply each maximum rate by the width of the interval:
Interval 1: 12.0 gal/hr * 2 hr = 24.0 gallons
Interval 2: 10.6 gal/hr * 2 hr = 21.2 gallons
Interval 3: 8.8 gal/hr * 2 hr = 17.6 gallons
Interval 4: 8.0 gal/hr * 2 hr = 16.0 gallons
Interval 5: 6.0 gal/hr * 2 hr = 12.0 gallons
Finally, we add up the upper estimates for each interval to find the total amount of water that leaked out:
Total upper estimate = 24.0 + 21.2 + 17.6 + 16.0 + 12.0 = 90.8 gallons
Therefore, the upper estimate for the total amount of water that leaked out using five rectangles is 90.8 gallons.