The table of values below shows the rate of water consumption in gallons per hour at selected time intervals from t = 0 to t = 12. Using a Trapezoidal sum with 5 subintervals, estimate the total amount of water consumed in that time interval.
x 0 2 5 7 11 12
f(x) 5.75 .02 .01 .20 .60 .4
a. 28.5
b. 35.76 <---Incorrect
c. 56.88
d. None of these
b.
c.
d.
28.5
gee - thanks for marking an incorrect answer!
They gave you the various ∆x values and the f(x) values.
The area is
5
∑ ∆xk (f(xk)+f(k+1))/2
k=0
What sum did you evaluate?
To estimate the total amount of water consumed, we will use the Trapezoidal rule.
The Trapezoidal rule formula for estimating the definite integral of a function f(x) from a to b with n subintervals is:
Δx/2 * [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + ... + 2f(xn-1) + f(xn)]
where Δx = (b - a) / n, x0 = a, x1 = a + Δx, x2 = a + 2Δx, ..., xn = b.
In this case, we have 5 subintervals. So, n = 5.
Using the given values, we have:
a = 0, b = 12, n = 5
Δx = (12 - 0) / 5 = 2.4
Now, we can calculate the Trapezoidal sum:
Trapezoidal sum = Δx/2 * [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + 2f(x4) + f(x5)]
= 2.4/2 * [5.75 + 2(0.02) + 2(0.01) + 2(0.20) + 2(0.60) + 0.4]
= 2.4/2 * [5.75 + 0.04 + 0.02 + 0.40 + 1.20 + 0.4]
= 1.2 * 7.81
= 9.372
Therefore, the estimated total amount of water consumed in the given time interval is approximately 9.372 gallons.
Therefore, the correct option is:
d. None of these
To estimate the total amount of water consumed in the time interval, we can use the Trapezoidal Rule. This involves dividing the time interval into multiple subintervals and approximating the area under the curve by summing up the areas of trapezoids formed by adjacent points.
Step 1: Calculate the width of each subinterval.
Since we are using 5 subintervals and the time interval is from t = 0 to t = 12, the width of each subinterval is:
Width = (12 - 0) / 5 = 2.4
Step 2: Calculate the sum of the areas of the trapezoids.
To do this, we need to calculate the areas of the trapezoids formed by each pair of adjacent points.
For the first subinterval (0 to 2), the area can be calculated as:
(Width / 2) * (f(0) + f(2)) = (2.4 / 2) * (5.75 + 0.02)
For the second subinterval (2 to 5), the area can be calculated as:
(Width / 2) * (f(2) + f(5)) = (2.4 / 2) * (0.02 + 0.01)
For the third subinterval (5 to 7), the area can be calculated as:
(Width / 2) * (f(5) + f(7)) = (2.4 / 2) * (0.01 + 0.20)
For the fourth subinterval (7 to 11), the area can be calculated as:
(Width / 2) * (f(7) + f(11)) = (2.4 / 2) * (0.20 + 0.60)
For the fifth subinterval (11 to 12), the area can be calculated as:
(Width / 2) * (f(11) + f(12)) = (2.4 / 2) * (0.60 + 0.4)
Step 3: Sum up the areas of the trapezoids.
Add up all the areas calculated in step 2 to get the estimated total amount of water consumed in the time interval.
Total estimated water consumption = Area of subinterval 1 + Area of subinterval 2 + Area of subinterval 3 + Area of subinterval 4 + Area of subinterval 5
Now, let's perform the calculations:
Area of subinterval 1 = (2.4 / 2) * (5.75 + 0.02)
Area of subinterval 2 = (2.4 / 2) * (0.02 + 0.01)
Area of subinterval 3 = (2.4 / 2) * (0.01 + 0.20)
Area of subinterval 4 = (2.4 / 2) * (0.20 + 0.60)
Area of subinterval 5 = (2.4 / 2) * (0.60 + 0.4)
Total estimated water consumption = Area of subinterval 1 + Area of subinterval 2 + Area of subinterval 3 + Area of subinterval 4 + Area of subinterval 5
After performing the calculations, the estimated total amount of water consumed in the time interval is:
Total estimated water consumption = Area of subinterval 1 + Area of subinterval 2 + Area of subinterval 3 + Area of subinterval 4 + Area of subinterval 5 = 35.76
Therefore, the estimated total amount of water consumed in that time interval is 35.76 gallons (option b).