To calculate the upper estimate for the total amount of water that leaked out of the tank using a Riemann sum with left endpoints, we follow these steps:
1. Identify the given data points:
- Time intervals (in hours): [tex]\( t = [0, 2, 4, 6, 8, 10] \)[/tex]
- Rate of water leakage (in gallons per hour) at given time intervals: [tex]\( r(t) = [8.7, 7.6, 6.8, 6.2, 5.7, 5.3] \)[/tex]
2. Determine the number of rectangles:
We use five rectangles as indicated in the problem. Therefore, the number of intervals is 5.
3. Calculate the width of each rectangle:
The width ([tex]\(\Delta t\)[/tex]) of each rectangle is the difference between consecutive time intervals. Here, [tex]\( t[1] - t[0] = 2 - 0 = 2 \)[/tex] hours.
4. Determine the heights using the left endpoints:
For the left endpoint Riemann sum, the heights of the rectangles will be the values of [tex]\( r(t) \)[/tex] at the left ends of each interval.
- For the interval [0, 2), the height is given by [tex]\( r(0) = 8.7 \)[/tex]
- For the interval [2, 4), the height is given by [tex]\( r(2) = 7.6 \)[/tex]
- For the interval [4, 6), the height is given by [tex]\( r(4) = 6.8 \)[/tex]
- For the interval [6, 8), the height is given by [tex]\( r(6) = 6.2 \)[/tex]
- For the interval [8, 10), the height is given by [tex]\( r(8) = 5.7 \)[/tex]
5. Calculate the area of each rectangle:
Since the width of each rectangle is 2, we can compute the area of each rectangle:
- [tex]\( 8.7 \times 2 = 17.4 \)[/tex]
- [tex]\( 7.6 \times 2 = 15.2 \)[/tex]
- [tex]\( 6.8 \times 2 = 13.6 \)[/tex]
- [tex]\( 6.2 \times 2 = 12.4 \)[/tex]
- [tex]\( 5.7 \times 2 = 11.4 \)[/tex]
6. Sum the areas to get the upper estimate:
The upper estimate is the sum of the areas of all rectangles:
[tex]\[
17.4 + 15.2 + 13.6 + 12.4 + 11.4 = 70.0
\][/tex]
Therefore, the upper estimate for the total amount of water that leaked out is [tex]\( 70.0 \)[/tex] gallons.
