The following table shows the rate of water flow (in L/min) through a dam.
t (min) 1 6 11 16 21 26 31
V'(t) (L/min) 1 8 2 9 7 5 9
Approximate the total volume of water that passed through the dam from t=1 to t=31 using Simpson's rule, with n=6.
15
To approximate the total volume of water that passed through the dam using Simpson's rule, we need to use the formula:
∫[a,b] f(x) dx ≈ (Δx/3) [f(a) + 4f(a+Δx) + 2f(a+2Δx) + 4f(a+3Δx) + ... + 4f(b-Δx) + f(b)]
where Δx = (b - a) / n
In this case, a = 1, b = 31, and n = 6. Let's calculate the approximate total volume step by step:
Step 1: Calculate Δx
Δx = (b - a) / n
Δx = (31 - 1) / 6
Δx = 5
Step 2: Calculate the sum of the terms in the formula
f(a) = V'(1) = 1
f(a+Δx) = V'(6) = 8
f(a+2Δx) = V'(11) = 2
f(a+3Δx) = V'(16) = 9
f(b-Δx) = V'(26) = 5
f(b) = V'(31) = 9
Sum = f(a) + 4f(a+Δx) + 2f(a+2Δx) + 4f(a+3Δx) + ... + 4f(b-Δx) + f(b)
= 1 + 4(8) + 2(2) + 4(9) + 4(5) + 9
= 1 + 32 + 4 + 36 + 20 + 9
= 102
Step 3: Calculate the approximate total volume
Total Volume ≈ (Δx/3) * Sum
= (5/3) * 102
≈ 170 L
Therefore, the approximate total volume of water that passed through the dam from t = 1 to t = 31 is approximately 170 liters using Simpson's rule.
To approximate the total volume of water that passed through the dam using Simpson's rule, we need to use the given data points and divide the interval from t=1 to t=31 into smaller subintervals.
Let's understand the steps to apply Simpson's rule:
Step 1: Calculate the step size (h) by dividing the total interval (t=31 - t=1) by the number of subintervals (n=6).
h = (31 - 1) / 6
= 5
Step 2: Calculate the sum of even and odd indexed terms.
- For the even indexed terms, add the values of V'(t) at t=6, 16, and 26. (V'(6) + V'(16) + V'(26))
- For the odd indexed terms, add the values of V'(t) at t=1, 11, 21, and 31. (V'(1) + V'(11) + V'(21) + V'(31))
Step 3: Multiply the sum of even indexed terms by 4 and the sum of odd indexed terms by 2.
- (4 * (V'(6) + V'(16) + V'(26))) + (2 * (V'(1) + V'(11) + V'(21) + V'(31)))
Step 4: Add the results of Step 3 to get the approximate total volume.
- Total Volume ≈ (h/3) * (Step 3 result)
Let's calculate:
Step 1: h = 5
Step 2: Sum of even indexed terms = V'(6) + V'(16) + V'(26)
= 8 + 9 + 5
= 22
Sum of odd indexed terms = V'(1) + V'(11) + V'(21) + V'(31)
= 1 + 2 + 7 + 9
= 19
Step 3: (4 * 22) + (2 * 19)
= 88 + 38
= 126
Step 4: Total Volume ≈ (5/3) * 126
≈ 210 L
Therefore, the approximate total volume of water that passed through the dam from t=1 to t=31 using Simpson's rule, with n=6, is 210 L.