Question

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.
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Answer 1

We can used the Simpson's Rule says to approximate the area under a given curve using the following formula: 

(Δx/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)] 

The pool is divided into 8 subintervals. We integrate the given function from 0 to 24, while the graph provides values of f(x) at 7 different points. The first value given, 6.2, is NOT f(0). It is f(3). Using Simpson's Rule, and dividing the lake of 24 meters into 8 subintervals, we write the equation: 

area = (3/3)[f(0) + 4f(3) + 2f(6) + 4f(9) +2f(12) + 4f(15) + 2f(18) + 4f(21) + f(24)] 

Pool area = 0 + 4(6.2) + 2(7.2) +4(6.8) + 2(5.6) + 4(5.0) +2(4.8) +4(4.8) + 0 = 126.4 m^2 

Rounding to the nearest square meter, the area of the lake is approximately 126 m^2 

Answer 2

Answer:

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