Electricity is consumed between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of consumption are shown in the table below, with t measured in hours past midnight, and R(t) measured in KW per hour. Use 4 trapezoids to estimate the total amount of electricity to the nearest KW consumed between midnight and 6 a.m.
t (hours) 0 1 2 5 6
R(t) KW/hr 400 200 350 200 150
I am wondering what the quesrion acrually is. Connect the points.
determine the area under the connected points. https://www.dummies.com/education/math/calculus/how-to-approximate-area-with-the-trapezoid-rule/
To estimate the total amount of electricity consumed between midnight (t = 0) and 6 a.m. (t = 6) using 4 trapezoids, we can use the Trapezoidal Rule of integration.
The Trapezoidal Rule approximates the area under a curve by dividing it into trapezoids, where the height of each trapezoid is the average of the two function values at the endpoints, and the width is the difference in x-values.
In this case, we have the following values:
t (hours): 0 1 2 3 4 5 6
R(t) KW/hr: 400 200 350 ?? ?? 200 150
We can see that we are missing two values for R(t) at t = 3 and t = 4. To estimate those values, we can take the average of the surrounding known values.
For t = 3:
R(3) = (R(2) + R(4)) / 2 = (350 + 200) / 2 = 275 KW/hr
For t = 4:
R(4) = (R(3) + R(5)) / 2 = (275 + 200) / 2 = 237.5 KW/hr
Now we have all the values we need to apply the Trapezoidal Rule. We divide the interval [0, 6] into four equal subintervals:
[0, 1], [1, 2], [2, 4], [4, 6]
The width of each subinterval is 6 / 4 = 1.5 hours.
Now, we calculate the areas of each trapezoid:
Area1 = (1.5 / 2) * (R(0) + R(1)) = (1.5 / 2) * (400 + 200) = 450 KW
Area2 = (1.5 / 2) * (R(1) + R(2)) = (1.5 / 2) * (200 + 350) = 412.5 KW
Area3 = (1.5 / 2) * (R(2) + R(4)) = (1.5 / 2) * (350 + 237.5) = 468.75 KW
Area4 = (1.5 / 2) * (R(4) + R(6)) = (1.5 / 2) * (237.5 + 150) = 262.5 KW
Finally, we sum up the areas of the four trapezoids:
Total area = Area1 + Area2 + Area3 + Area4 = 450 + 412.5 + 468.75 + 262.5 = 1593.75 KW
Therefore, the estimated total amount of electricity consumed between midnight and 6 a.m. is approximately 1594 KW.