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 = 15.5, a2 = 18, a3 = 17, a4 = 14, a5 = 12.5, a6 = 12, and a7 = 12.

To estimate the area of the kidney-shaped swimming pool using the Midpoint Rule, we need to calculate the width of each interval.

First, let's find the midpoint of each interval by adding the widths of adjacent points and dividing by 2:

a1 = 15.5
a2 = 18
a3 = 17
a4 = 14
a5 = 12.5
a6 = 12
a7 = 12

Now, calculate the width of each interval by subtracting the adjacent midpoints:

w1 = a2 - a1 = 18 - 15.5 = 2.5
w2 = a3 - a2 = 17 - 18 = -1
w3 = a4 - a3 = 14 - 17 = -3
w4 = a5 - a4 = 12.5 - 14 = -1.5
w5 = a6 - a5 = 12 - 12.5 = -0.5
w6 = a7 - a6 = 12 - 12 = 0

Notice that some intervals have negative widths. This indicates that the swimming pool is concave at those intervals. We'll consider both positive and negative widths in our area estimation.

Now, let's calculate the estimated area using the Midpoint Rule formula:

S ≈ (w1*f(x1) + w2*f(x2) + w3*f(x3) + w4*f(x4)) / 2

where f(x) is the function that defines the shape of the pool.

In this case, since the pool is kidney-shaped, we can represent it as two half-ellipses placed back to back. The equation of an ellipse can be approximated by:

A = π * a * b

where a is the semi-major axis and b is the semi-minor axis.

Let's assume the swimming pool can be approximated as an ellipse with a = 18 and b = 12.

Now we can calculate the estimated area:

S ≈ (w1*A1 + w2*A2 + w3*A3 + w4*A4) / 2

Substituting the values we have:

S ≈ (2.5 * π * 18 * 12 + (-1) * π * 17 * 12 + (-3) * π * 14 * 12 + (-1.5) * π * 12.5 * 12) / 2

Now we can calculate this expression to find the estimated area S.

To estimate the area of the kidney-shaped swimming pool using the Midpoint Rule with n = 4, we can approximate the shape of the pool by dividing it into rectangles and calculating the total area of those rectangles.

First, we need to find the intervals on which to divide the pool. Since we have measurements at 3-meter intervals, we can divide the pool into 6 intervals of width 3 meters.

Next, we calculate the midpoint of each interval. The midpoints for the given widths are as follows:

Interval 1: (15.5 + 18) / 2 = 16.75
Interval 2: (18 + 17) / 2 = 17.5
Interval 3: (17 + 14) / 2 = 15.5
Interval 4: (14 + 12.5) / 2 = 13.25
Interval 5: (12.5 + 12) / 2 = 12.25
Interval 6: (12 + 12) / 2 = 12

Now we can calculate the area of each rectangle. The height of each rectangle is 3 meters since that is the width of the intervals.

Rectangle 1: 3 * 16.75 = 50.25 square meters
Rectangle 2: 3 * 17.5 = 52.5 square meters
Rectangle 3: 3 * 15.5 = 46.5 square meters
Rectangle 4: 3 * 13.25 = 39.75 square meters
Rectangle 5: 3 * 12.25 = 36.75 square meters
Rectangle 6: 3 * 12 = 36 square meters

Finally, we add up the areas of all the rectangles to get the estimated area of the pool:

S = 50.25 + 52.5 + 46.5 + 39.75 + 36.75 + 36 = 261.75 square meters

Therefore, the estimated area of the kidney-shaped swimming pool using the Midpoint Rule with n = 4 is 261.75 square meters.

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