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
did you draw the graph? The area of a trapezoid is (B+b)/2 * h
so, the amount (area) will be
(400+200)/2*(1-0) + (200+350)/2*(2-1) + (200+350)/2*(5-2) + (150+200)/2*(6-5) = 1575
Well, I hope you're ready for a shocking estimation! Let's calculate the total amount of electricity consumed between midnight and 6 a.m. using trapezoids.
The trapezoidal rule is a method of numerical integration, and in this case, we can use it to estimate the area under the rate of consumption curve.
First, let's calculate the width of each trapezoid. Since we have 4 trapezoids, the width of each will be (6-0)/4 = 1.5 hours.
Now, let's calculate the area of each trapezoid. The formula for the area of a trapezoid is (base1 + base2) * height / 2.
For the first trapezoid: base1 = R(0) = 400 KW/hr, base2 = R(1) = 200 KW/hr, height = 1.5 hours.
Area1 = (400 + 200) * 1.5 / 2 = 450 KW.
For the second trapezoid: base1 = R(1) = 200 KW/hr, base2 = R(2) = 350 KW/hr, height = 1.5 hours.
Area2 = (200 + 350) * 1.5 / 2 = 412.5 KW.
For the third trapezoid: base1 = R(2) = 350 KW/hr, base2 = R(5) = 200 KW/hr, height = 1.5 hours.
Area3 = (350 + 200) * 1.5 / 2 = 412.5 KW.
For the fourth trapezoid: base1 = R(5) = 200 KW/hr, base2 = R(6) = 150 KW/hr, height = 1.5 hours.
Area4 = (200 + 150) * 1.5 / 2 = 262.5 KW.
Now, let's add up the areas of all four trapezoids to get the total estimated amount of electricity consumed:
Total estimated electricity consumed = Area1 + Area2 + Area3 + Area4
Total estimated electricity consumed = 450 + 412.5 + 412.5 + 262.5 = 1537.5 KW.
So, according to my electrifying calculations, the total estimated amount of electricity consumed between midnight and 6 a.m. is 1537.5 KW.
To estimate the total amount of electricity consumed between midnight and 6 a.m. using 4 trapezoids, we can use the formula for the area of a trapezoid:
Area = (b1 + b2) * h / 2
where b1 and b2 are the lengths of the parallel sides of the trapezoid and h is the height.
In this case, the length of each base (b1 and b2) will be the time difference between two adjacent data points, and the height (h) will be the average of the two corresponding consumption rates.
Let's calculate the area for each trapezoid and then sum them up to get the total amount of electricity consumed.
Trapezoid 1:
Base 1 = t2 - t0 = 1 - 0 = 1 hour
Base 2 = t2 - t1 = 1 - 0 = 1 hour
Height = (R(t0) + R(t1)) / 2 = (400 + 200) / 2 = 300 KW/hr
Area1 = (1 + 1) * 300 / 2 = 600 / 2 = 300 KW
Trapezoid 2:
Base 1 = t5 - t2 = 5 - 2 = 3 hours
Base 2 = t5 - t2 = 5 - 2 = 3 hours
Height = (R(t2) + R(t5)) / 2 = (350 + 200) / 2 = 275 KW/hr
Area2 = (3 + 3) * 275 / 2 = 1650 / 2 = 825 KW
Trapezoid 3:
Base 1 = t6 - t5 = 6 - 5 = 1 hour
Base 2 = t6 - t5 = 6 - 5 = 1 hour
Height = (R(t5) + R(t6)) / 2 = (200 + 150) / 2 = 175 KW/hr
Area3 = (1 + 1) * 175 / 2 = 350 / 2 = 175 KW
Total Area = Area1 + Area2 + Area3 = 300 + 825 + 175 = 1300 KW
Therefore, the total amount of electricity consumed between midnight and 6 a.m. is approximately 1300 KW.
To estimate the total amount of electricity consumed between midnight and 6 a.m. using four trapezoids, we can use the trapezoidal rule for approximating an integral.
The formula for the trapezoidal rule is:
∫[a, b] f(x) dx ≈ ((b-a)/2n) * (f(a) + 2f(a+h) + 2f(a+2h) + ... + 2f(b-h) + f(b))
In this case, a = 0 (midnight) and b = 6 a.m. We have four trapezoids, so n = 4.
First, we need to calculate the width of each trapezoid, which is (b-a)/n:
h = (6-0)/4 = 1.5
Now, we can calculate the approximate integral using the trapezoidal rule:
∫[0, 6] R(t) dt ≈ (1.5/2) * (R(0) + 2R(1.5) + 2R(3) + 2R(4.5) + R(6))
Substituting the given values of R(t) into the formula:
≈ (1.5/2) * (400 + 2*200 + 2*350 + 2*200 + 150)
Calculating:
≈ (0.75) * (400 + 400 + 700 + 400 + 150)
≈ (0.75) * (2050)
≈ 1537.5
Therefore, the estimated total amount of electricity consumed between midnight and 6 a.m. to the nearest KW is 1538 KW.