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.

Question

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

Answer 1

To estimate the total amount of electricity consumed between midnight and 6 a.m. using 4 trapezoids, we can approximate the integral of the rate of consumption function R(t) over the interval [0, 6].

First, let's divide the interval [0, 6] into 4 subintervals of equal width. Each subinterval will have a width of (6-0)/4 = 1.5 hours.

Next, let's calculate the average rate of consumption, denoted as R_avg, for each subinterval by taking the average of the rates at the endpoints of the subinterval.

For the first subinterval [0, 1.5]:
R_avg = (R(0) + R(1.5))/2 = (400 + 200)/2 = 300 KW/hr

For the second subinterval [1.5, 3]:
R_avg = (R(1.5) + R(3))/2 = (200 + 350)/2 = 275 KW/hr

For the third subinterval [3, 4.5]:
R_avg = (R(3) + R(4.5))/2 = (350 + 200)/2 = 275 KW/hr

For the fourth subinterval [4.5, 6]:
R_avg = (R(4.5) + R(6))/2 = (200 + 150)/2 = 175 KW/hr

Now, let's calculate the area of each trapezoid, which represents the approximate contribution of each subinterval to the total amount of consumed electricity. The area of a trapezoid is given by the formula:

Area = (b1 + b2) * h / 2,

where b1 and b2 are the lengths of the parallel sides (average rates of consumption) and h is the height (width of the subinterval).

For the first trapezoid:
Area1 = (R_avg1 + R_avg2) * h / 2 = (300 + 275) * 1.5 / 2 = 243.75 KW

For the second trapezoid:
Area2 = (R_avg2 + R_avg3) * h / 2 = (275 + 275) * 1.5 / 2 = 412.50 KW

For the third trapezoid:
Area3 = (R_avg3 + R_avg4) * h / 2 = (275 + 175) * 1.5 / 2 = 337.50 KW

For the fourth trapezoid:
Area4 = (R_avg4 + R_avg5) * h / 2 = (175 + 150) * 1.5 / 2 = 243.75 KW

Finally, to estimate the total amount of electricity consumed, we sum up the areas of all four trapezoids:
Total Electricity Consumed = Area1 + Area2 + Area3 + Area4 = 243.75 + 412.50 + 337.50 + 243.75 = 1237.50 KW

Therefore, the estimated total amount of electricity consumed between midnight and 6 a.m. is 1238 KW (rounded to the nearest KW).