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.
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.

I am trying and don't think i am getting the right answer.

well, you could choose a left Riemann sum or a right Riemann sum or an in between Riemann sum. For simplicity I will take the left

each day is 24 hours * 30 days = 720 hours
so
area = kw hr = p(0) + p(1) + p(2) ...+ p(719) [ each times one hour ]

B)
KwHr = integral from t = 0 to t = 720 of f(t) dt

your table I assume should show f(t) not f(x)
area = 2.3*1 + 2.5* 1 + 2.1*1 etc to 1.0 *1
because I am using the left sum option, I am not using (10, 1.8)
so add the first ten numbers to get your answer
I get 31.7 kw hr
which is about an average power of 3.17 kw during the ten hours.
It would be more accurate to then do a right sum using 1 to 10 and not using (0,2.3) then use the average of the right method and left method.

Well, it sounds like you're really "powering" through these calculations! Let's break it down together.

A. To approximate the total number of kilowatt-hours used by the customer in the 30-day period using a Riemann sum, you need to divide the period into smaller intervals, calculate the power usage in each interval, and add them up.

Let's start by dividing the 30-day period into smaller intervals, let's say n intervals. Then, the width of each interval would be (30 * 24) hours / n. Next, we approximate the power usage in each interval by evaluating P(t) at some point within each interval. Multiply this power by the width of each interval, and keep summing up these products for all the intervals. As n approaches infinity, this sum will converge to the exact value of kilowatt-hours used in the 30-day period.

B. To derive an expression representing the total number of kilowatt-hours used by the customer in the 30-day period, you need to integrate the function P(t) over the interval [0, 30 * 24] (since there are 30 days with 24 hours each). The integral will give you the exact value of the kilowatt-hours used.

C. Now, let's tackle the given data for the function. From the data, we can approximate the power usage at each time point by taking the average of the two neighboring values. For example, between (0,2.3) and (1,2.5), we can approximate the power usage as (2.3 + 2.5)/2 = 2.4 kilowatts. Next, multiply each approximate power usage by the time interval (which is 1 hour in this case) to get an estimation of kilowatt-hours used in each interval. Finally, add up these estimated kilowatt-hours for the desired 10-hour period.

I hope this helps you out! If you need any more "watt"-tastic explanations, feel free to ask!

To approximate the total number of kilowatt-hours used by the customer in the 30-day period using a Riemann sum, you can divide the 30-day period into smaller intervals and calculate the energy consumption for each interval.

A.

Let's assume that the 30-day period is divided into n intervals of equal length. The length of each interval would be 30 days divided by n. Let's denote this length as Δt.

Now, we need to find the amount of power used during each interval. We can approximate this by taking the average power over the interval. Suppose the interval starts at time t_i and ends at time t_i+1. The average power over this interval is given by:

P_avg = (P(t_i) + P(t_i+1))/2.

To calculate the energy consumption for each interval, we need to multiply the average power by the length of the interval:

ΔE = P_avg * Δt.

Summing up the energy consumption over all intervals will give us an approximation of the total number of kilowatt-hours used by the customer in the 30-day period:

Total energy ≈ Σ(ΔE) = Σ(P_avg * Δt).

To get a better approximation, you can choose a larger value of n and repeat the above steps.

B.

To derive an expression representing the total number of kilowatt-hours used by the customer in the 30-day period, we need to take the limit as n approaches infinity of our Riemann sum approximation. This will give us the exact value of the integral:

Total energy = ∫(P(t))dt.

This expression represents the accumulated energy consumption over the entire 30-day period.

C.

To estimate the number of kilowatt-hours used by the customer during the 10-hour period using the given data, you can use the trapezoidal rule.

First, calculate the average power for each interval using the given data:

Average power for first interval = (2.3 + 2.5) / 2 = 2.4
Average power for second interval = (2.5 + 2.1) / 2 = 2.3
Average power for third interval = (2.1 + 3.9) / 2 = 3.0
Average power for fourth interval = (3.9 + 3.6) / 2 = 3.75
Average power for fifth interval = (3.6 + 5.5) / 2 = 4.55
Average power for sixth interval = (5.5 + 4.5) / 2 = 5.0
Average power for seventh interval = (4.5 + 5.6) / 2 = 5.05
Average power for eighth interval = (5.6 + 1.2) / 2 = 3.4
Average power for ninth interval = (1.2 + 1.0) / 2 = 1.1
Average power for tenth interval = (1.0 + 1.8) / 2 = 1.4

Now, calculate the energy consumption for each interval by multiplying the average power by the length of the interval:

Energy for first interval = 2.4 * 1 = 2.4 kilowatt-hours
Energy for second interval = 2.3 * 1 = 2.3 kilowatt-hours
Energy for third interval = 3.0 * 1 = 3.0 kilowatt-hours
Energy for fourth interval = 3.75 * 1 = 3.75 kilowatt-hours
Energy for fifth interval = 4.55 * 1 = 4.55 kilowatt-hours
Energy for sixth interval = 5.0 * 1 = 5.0 kilowatt-hours
Energy for seventh interval = 5.05 * 1 = 5.05 kilowatt-hours
Energy for eighth interval = 3.4 * 1 = 3.4 kilowatt-hours
Energy for ninth interval = 1.1 * 1 = 1.1 kilowatt-hours
Energy for tenth interval = 1.4 * 1 = 1.4 kilowatt-hours

Finally, sum up the energy consumption for all intervals to get the estimate for the 10-hour period:

Total energy ≈ 2.4 + 2.3 + 3.0 + 3.75 + 4.55 + 5.0 + 5.05 + 3.4 + 1.1 + 1.4 = 32.95 kilowatt-hours.

Therefore, it is estimated that the customer used approximately 32.95 kilowatt-hours in the 10-hour period based on the given data.

To approximate the total number of kilowatt-hours used by the customer over the 30-day period, we can use a Riemann sum. A Riemann sum is an approximation of the area under a curve by dividing it into smaller rectangles and summing their areas.

A. To approximate the total number of kilowatt-hours used, we can use a left Riemann sum. This means that we will use the value of the function at the left endpoint of each subinterval to calculate the height of each rectangle.

Here's how to calculate the Riemann sum:

1. Divide the 30-day period into smaller intervals or subintervals. Let's say we choose n subintervals.
2. Determine the width of each subinterval by dividing the total time (30 days) by the number of subintervals: Δt = (30 days) / n.
3. Choose a value of t from each subinterval as the left endpoint. For example, if n = 5, the left endpoints could be 0, 6, 12, 18, and 24.
4. Evaluate the function P(t) at each left endpoint to find the height of each rectangle: P(t_1), P(t_2), ..., P(t_n).
5. Calculate the area of each rectangle by multiplying the height (P(t)) by the width (Δt): P(t_1) * Δt, P(t_2) * Δt, ..., P(t_n) * Δt.
6. Sum up the areas of all the rectangles to approximate the total kilowatt-hours used: P(t_1) * Δt + P(t_2) * Δt + ... + P(t_n) * Δt.

B. To derive an expression representing the total number of kilowatt-hours used by the customer in the 30-day period, we need to take the limit as the number of subintervals approaches infinity. This gives us the definite integral of P(t) over the interval [0, 30 days]:

Total kilowatt-hours = ∫[0, 30 days] P(t) dt

C. To estimate the number of kilowatt-hours the customer uses in the 10-hour period using the given data, we can calculate the Riemann sum using the provided data points.

1. Divide the 10-hour period into smaller intervals or subintervals. Let's say we choose n = 10 subintervals.
2. Determine the width of each subinterval by dividing the total time (10 hours) by the number of subintervals: Δt = (10 hours) / n = 1 hour.
3. Evaluate the function f(t) at each left endpoint of the subintervals using the given data points: f(0), f(1), f(2), ..., f(10).
4. Calculate the area of each rectangle by multiplying the height (f(t)) by the width (Δt): f(0) * Δt, f(1) * Δt, ..., f(10) * Δt.
5. Sum up the areas of all the rectangles to estimate the total kilowatt-hours used in the 10-hour period: f(0) * Δt + f(1) * Δt + ... + f(10) * Δt.

By following these steps, you should be able to approximate the kilowatt-hours used by the customer in the given time periods and derive an expression for the total kilowatt-hours used over a 30-day period.

By the way

POWER IS NOT ENERGY

look in physics book NEVER in math book ! (mathematics text writers do not get this stuff. They do not know the difference between bearing and heading in navigation either)

Energy = power * time

in this problem
Energy is in KW HR
and Power is in KW

This sentence should be:

A kilowatt is a measure of how much power (******* leave that energy word OUT ***** ) a customer is using ...........