Power companies typically bill customers based on the number of kilowatt-hours used during a single billing period. A kilowatt is a measure of how much power (energy) a
customer is using, while a kilowatt-hour is one kilowatt of power being used for one hour.
For constant power use, the number of kilowatt-hours used is calculated by kilowatt-hours=kilowatts * time (in hours). Thus, if customers use 5 kilowatts for 30 minutes, they'll have used 5 kilowatts * (1/2)hrs =2.5 kilowatt-hours.
Suppose the power use of a customer over a 30-day period is given by the continuous
function P(t) where P is kilowatts, t is time in hours, and t =0 corresponds to the
beginning of the 30 day period.
A.
Approximate, with a Riemann sum, the total number of kilowatt-hours used by the customer in the 30 days. Please tell sigma notation, and exact approximation.
B.
Derive an expression representing the total number of kilowatt-hours used by the
customer in the 30-day period. (This expression should not be an approximation.)
C. Consider the following data for the function.
t f(x)
0 2.3
1 2.5
2 2.1
3 3.9
4 3.6
5 5.5
6 4.5
7 5.6
8 1.2
9 1.0
10 1.8
Recall that f(t) represents the number of kilowatts being used by a customer at time t hours from the beginning of the billing period. Estimate the number of kilowatt-hours the customer uses in this 10-hour period, and explain your method.
B. 24*30 = 720 hours so the energy used is
∫[0,720] P(t) dt
A. You can approximate that using n intervals, each of width h = 720/n
by the Right-hand sum
n
∑ P(720/n) * h
k=1
C. Use n=10
Please elaborate B and C.
A. To approximate the total number of kilowatt-hours used by the customer in the 30-day period using a Riemann sum, we need to divide the time range into smaller intervals. Let's say we divide the 30-day period into n equal intervals, which means each interval has a duration of 30 * 24 / n hours.
The Riemann sum formula for approximating the total number of kilowatt-hours is:
ΣP(ti) * Δt
Here, P(ti) represents the power usage at time ti, and Δt is the duration of each interval.
Using sigma notation, the Riemann sum can be written as:
Σ[ P(ti) * Δt ] for i = 1 to n
To find the exact approximation, we need to take the limit of this Riemann sum as n approaches infinity:
lim(n→∞) Σ[ P(ti) * Δt ] for i = 1 to n
B. To derive an expression representing the total number of kilowatt-hours used by the customer in the 30-day period, we can integrate the function P(t) over the interval [0, 30 * 24] (since the 30-day period is equivalent to 30 * 24 hours).
The expression for the total number of kilowatt-hours is:
∫[0, 30 * 24] P(t) dt
C. To estimate the number of kilowatt-hours used by the customer in the 10-hour period using the given data, we can use the trapezoidal rule. The trapezoidal rule estimates the integral by approximating the area under the curve using trapezoids.
The formula for the estimation using the trapezoidal rule is:
Approximation = (h/2) * [f(t1) + 2f(t2) + 2f(t3) + ... + 2f(tn-1) + f(tn)]
Here, h is the width of each interval, and f(ti) represents the value of P(t) at ti.
Using the given data:
t f(t)
0 2.3
1 2.5
2 2.1
3 3.9
4 3.6
5 5.5
6 4.5
7 5.6
8 1.2
9 1.0
10 1.8
We can divide the 10-hour period into 10 equal intervals, so h = 1. Then, we can calculate the approximation as follows:
Approximation = (1/2) * [2.3 + 2 * (2.5 + 2.1 + 3.9 + 3.6 + 5.5 + 4.5 + 5.6 + 1.2 + 1.0) + 1.8]
Simplifying the expression gives us the estimated number of kilowatt-hours used by the customer in the 10-hour period.