Question

The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson’s Rule to estimate the area of the pool.

Answer 1

To estimate the area of the kidney-shaped swimming pool using Simpson's Rule, we will divide the pool into several sections and approximate each section using a quadratic function.

First, we have the width measurements at 2-meter intervals:

Widths (in meters):
2, 4, 6, 8, 10, 12, 14, 12, 10, 8, 6, 4, 2

Now, we will divide these measurements into pairs to apply Simpson's Rule. Since we have 13 width measurements, we will have 6 intervals.

Interval 1: (2, 4, 6)
Interval 2: (6, 8, 10)
Interval 3: (10, 12, 14)
Interval 4: (14, 12, 10)
Interval 5: (10, 8, 6)
Interval 6: (6, 4, 2)

Next, we will use Simpson's Rule to estimate the area of each interval. The formula for Simpson's Rule is:

Area ≈ (Δx/3) [y0 + 4y1 + 2y2 + 4y3 + ... + 2yn-2 + 4yn-1 + yn]

For each interval, the Δx = 2 (since the width measurements were taken at 2-meter intervals).

Calculating the estimated area for each interval:

Interval 1: Area ≈ (2/3) [6 + 4(8) + 2(10)] ≈ 32
Interval 2: Area ≈ (2/3) [10 + 4(12) + 2(14)] ≈ 56
Interval 3: Area ≈ (2/3) [14 + 4(12) + 2(10)] ≈ 52
Interval 4: Area ≈ (2/3) [10 + 4(8) + 2(6)] ≈ 32
Interval 5: Area ≈ (2/3) [6 + 4(4) + 2(2)] ≈ 16
Interval 6: Area ≈ (2/3) [2 + 4(4) + 2(6)] ≈ 20

Finally, we add up the areas of each interval to get the estimated total area of the pool:

Total Estimated Area ≈ 32 + 56 + 52 + 32 + 16 + 20 ≈ 208 square meters

Therefore, the estimated area of the kidney-shaped swimming pool using Simpson's Rule is approximately 208 square meters.