The measurements of a meadow were taken at 30 ft intervals. Use Simpson's Rule to estimate the area of the meadow with the following measurements: 76 ft, 118 ft, 130 ft, 143 ft, 139 ft, 136 ft, 137 ft, 139 ft, 130 ft, 122 ft, 60 ft

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

= 30/3 * (76 + 4*118 + 2*130 + 4*143 + 2*139 + 4*136 + 2*137 + 4*139 + 2*130 + 4*122 + 60)

To estimate the area of the meadow using Simpson's Rule, follow these steps:

1. Determine the number of data points. In this case, we have 11 data points.

2. Calculate the step size (h) by dividing the total range of the data by the number of intervals. In this case, the range of the data is 76 ft to 143 ft, so the total range is 143 ft - 76 ft = 67 ft. Dividing this by the number of intervals (11 - 1 = 10), we get h = 67 ft / 10 = 6.7 ft.

3. Calculate the sum of the first and last data points, each multiplied by 1. In this case, the first data point is 76 ft and the last data point is 60 ft, so the sum is 76 ft + 60 ft = 136 ft.

4. Calculate the sum of the remaining data points, each multiplied by either 2 or 4, alternating. In this case, we have: 118 ft * 4 + 130 ft * 2 + 143 ft * 4 + 139 ft * 2 + 136 ft * 4 + 137 ft * 2 + 139 ft * 4 + 130 ft * 2 + 122 ft * 4 = 472 ft + 260 ft + 572 ft + 278 ft + 544 ft + 274 ft + 556 ft + 260 ft + 488 ft = 3414 ft.

5. Multiply the sum from step 4 by the step size divided by 3. In this case, 3414 ft * (6.7 ft / 3) = 3414 ft * 2.23 ft ≈ 7615.2 ft².

Therefore, the estimated area of the meadow is approximately 7615.2 square feet.

To estimate the area of the meadow using Simpson's Rule, we first need to divide the measurements into equal subintervals.

Simpson's Rule is a numerical approximation method that uses a quadratic equation to estimate the area under a curve. It works by approximating the curve as a series of parabolic segments, fitting these segments to the data points, and then integrating each segment to find the area.

Step 1: Group the measurements into subintervals
Divide the measurements into pairs, starting with the first measurement, then skipping one, and so on. In this case, we have 11 measurements, so we will have 5 subintervals:
(76 ft, 118 ft), (130 ft, 143 ft), (139 ft, 136 ft), (137 ft, 139 ft), (130 ft, 122 ft)

Step 2: Calculate the width of each subinterval
The width of each subinterval is simply the difference between the x-values (measurements) of each pair. Calculate the width for each subinterval:
Interval 1 width = 118 ft - 76 ft = 42 ft
Interval 2 width = 143 ft - 130 ft = 13 ft
Interval 3 width = 136 ft - 139 ft = -3 ft (Note: The value is negative in this case, which will be addressed later)
Interval 4 width = 139 ft - 137 ft = 2 ft
Interval 5 width = 122 ft - 130 ft = -8 ft

Step 3: Calculate the sum of the y-values for each subinterval
In this case, the y-values correspond to the width of each subinterval:
Interval 1 sum of y-values = 42 ft
Interval 2 sum of y-values = 13 ft
Interval 3 sum of y-values = -3 ft
Interval 4 sum of y-values = 2 ft
Interval 5 sum of y-values = -8 ft

Step 4: Apply Simpson's Rule formula
The area estimate using Simpson's Rule is given by:
Area ≈ (width/3) * (y1 + 4y2 + 2y3 + 4y4 + 2y5 + yn)
where n is the number of subintervals.

Applying the formula, we get:
Area ≈ (42/3) * (42 ft + 4(13 ft) + 2(-3 ft) + 4(2 ft) + 2(-8 ft))
Area ≈ (14) * (42 ft + 52 ft - 6 ft + 8 ft - 16 ft)
Area ≈ 196 ft * 80 ft
Area ≈ 15,680 ft²

So, the estimated area of the meadow using Simpson's Rule with the given measurements is approximately 15,680 square feet.