The area a meadow was approximated by measuring the length of the meadow at 30-foot intervals. THe distances measured across the meadow were 76 ft, 118 ft, 130 ft, 143 ft, 139 ft, 136 ft, 137 ft, 139 ft, 130 ft, 122 ft, and 60 ft. Use Simpson's Rule to approximate the area of the meadow.
Have you run out of names yet?
Simpson's rule is easy, accurate and fun to apply. Give it a try. I assume you know the formula.
(h/3)(x1 + 4x2 + 2x3 +... 4xn-1 + xn)
There have to be an odd number of measurement points, and that is the case here.
h = 30m in this case.
the x's are the widths at each y position. x1 = 76 etc.
I guess you have Mrs. Pompelia
To approximate the area of the meadow using Simpson's Rule, we first need to know the intervals between the measured distances. In this case, the intervals are 30 feet apart.
Simpson's Rule provides an approximate value for the area under a curve by using quadratic interpolation. It uses the formula:
Area ≈ (1/3) * h * [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_n-2) + 4f(x_n-1) + f(x_n)],
where h is the interval length and f(x) represents the measured distances.
Let's start by listing the measured distances:
76 ft, 118 ft, 130 ft, 143 ft, 139 ft, 136 ft, 137 ft, 139 ft, 130 ft, 122 ft, and 60 ft.
Since the intervals are 30 feet apart, we can assign x-values to each measurement:
x_0 = 0, x_1 = 30, x_2 = 60, x_3 = 90, x_4 = 120, x_5 = 150, x_6 = 180, x_7 = 210, x_8 = 240, x_9 = 270, x_10 = 300.
Now, we can substitute these values into the Simpson's Rule formula. Since there are 11 measured distances, the summation will go up to n=10:
Area ≈ (1/3) * 30 * [f(0) + 4f(30) + 2f(60) + 4f(90) + 2f(120) + 4f(150) + 2f(180) + 4f(210) + 2f(240) + 4f(270) + f(300)].
Plugging in the measured distances, the formula becomes:
Area ≈ (1/3) * 30 * [76 + 4 * 118 + 2 * 130 + 4 * 143 + 2 * 139 + 4 * 136 + 2 * 137 + 4 * 139 + 2 * 130 + 4 * 122 + 60].
Calculating this expression will give you an approximation of the area of the meadow using Simpson's Rule.