The widths (in meters) of a kidney-shaped swimming pool were measured at 3-meter intervals as indicated in the figure. use the midpoint rule with n = 4 to estimate the area s of the pool if a1 = 9.3, a2 = 10.8, a3 = 10.2, a4 = 8.4, a5 = 7.5, a6 = 7.2, and a7 = 7.2.

Area of the pool = 177.6 m^2

Step-by-step explanation:

Every measure in the pool can be seen as a value of a function f(x)

The distance between each measure is represented in variable x

The pool is 24 m long (8 intervals of 3 meters each)

At the beginning and the end of the pool (x = 0 and x = 24) the function is equal to zero

n = 4 means that we have to use only 4 intervals. Using points a2, a4, a6 and a8 we get

x    | 0   | 6     | 12   | 18   | 24 |

f(x) | 0   | 10.8 | 8.4 | 7.2 | 0   |  

The simpson's rule states that the integral of the function, i.e. the are S of the pool,  can be approximated as follows:

Area = (Δx/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + f(x4)]  

Replacing with Δx =  6 and the values of the function we get:

Area = (6/3)[0+ 4(10.8) + 2(8.4) + 4(7.2) + 0]   = 177.6 m^2

step-by-step explanation:

1. 3.4/4

2. 0.8/7

3. 0.5 x 3

4. 0.9 x 6

5. 10.8

6. 2.35

7. 0.567

8. 9.35

9. 0.02

10. 0.06

11. 0.16

12. 0.42

13. 0.63

14. 5.6

15. 8.1

16. 9.9

17. 0.28

18. 0.48

19. 0.27

20. 0.49

21. 0.45

22. 0.72

23. 0.9

24. 0.88

25. 0.75

26. 0.16

27. 0.328

28. 0.495

29. 0.72

30. 1.068