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.
t (hr) 0 2 4 6 8 10
r(t) (gal/hr) 8.7 7.6 6.8 6.2 5.7 5.3
since the width of each rectangle is 2, the sum is just
2(f(0)+f(2)+...+f(8))
since we're using left-side values.
To find the upper estimate for the total amount of water that leaked out using five rectangles, we need to calculate the area of each rectangle and then sum them up.
First, we need to determine the width of each rectangle. Since the time intervals are 2 hours apart, the width of each rectangle will be 2 hours.
Next, we need to determine the height of each rectangle. The height of each rectangle will be the maximum value of the rate within the time interval it represents.
Given the table, we can see that:
- The first rectangle represents the time interval from 0 to 2 hours, with a rate of 8.7 gallons per hour. Therefore, the height of the first rectangle is 8.7 gallons per hour.
- Similarly, the second rectangle represents the time interval from 2 to 4 hours, with a rate of 7.6 gallons per hour. The height of the second rectangle is 7.6 gallons per hour.
- Continuing this pattern, the heights of the remaining rectangles are: 6.8, 6.2, and 5.7 gallons per hour.
Now, let's calculate the area of each rectangle and sum them up:
Area of rectangle 1 = width * height = 2 hours * 8.7 gallons per hour = 17.4 gallons
Area of rectangle 2 = 2 hours * 7.6 gallons per hour = 15.2 gallons
Area of rectangle 3 = 2 hours * 6.8 gallons per hour = 13.6 gallons
Area of rectangle 4 = 2 hours * 6.2 gallons per hour = 12.4 gallons
Area of rectangle 5 = 2 hours * 5.7 gallons per hour = 11.4 gallons
Finally, summing up the areas of all five rectangles:
Total amount of water leaked out (upper estimate) = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4 + Area of rectangle 5
= 17.4 gallons + 15.2 gallons + 13.6 gallons + 12.4 gallons + 11.4 gallons
= 70 gallons
Therefore, the upper estimate for the total amount of water that leaked out using five rectangles is 70 gallons.
To find the upper estimate for the total amount of water that leaked out using five rectangles, we will use the left end-points of each rectangle.
Step 1: Calculate the width of each rectangle.
The width is equal to the time interval between each two-hour period.
Width = 2 hours
Step 2: Calculate the area of each rectangle.
The area of each rectangle is equal to the rate of water leakage multiplied by the width.
Rectangle 1:
Area = rate * width = 8.7 * 2 = 17.4 gallons
Rectangle 2:
Area = rate * width = 7.6 * 2 = 15.2 gallons
Rectangle 3:
Area = rate * width = 6.8 * 2 = 13.6 gallons
Rectangle 4:
Area = rate * width = 6.2 * 2 = 12.4 gallons
Rectangle 5:
Area = rate * width = 5.7 * 2 = 11.4 gallons
Step 3: Sum up the areas of all five rectangles to find the upper estimate for the total amount of water leaked.
Upper estimate = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4 + Area of rectangle 5
Upper estimate = 17.4 + 15.2 + 13.6 + 12.4 + 11.4 = 70 gallons
Therefore, the upper estimate for the total amount of water that leaked out using five rectangles is 70 gallons.