Question

Home Electricity Use The rate at which your home consumes electricity is measured in kilowatts. If your home consumes electricity at the rate of 1 kilowatt for 1 hour, you will be charged for 1 "kilowatt-hour" of electricity. Suppose that the average consumption rate for a certain home is modeled by the function \(C(t)=3.9-2.4 \sin (\pi t / 12),\) where \(C(t)\) is measured in kilowatts and \(t\) is the number of hours past midnight. Find the average daily consumption for this home, measured in kilowatt- hours.

Short answer

To get the average daily consumption in kilowatt-hours, integrate the given function from 0 to 24. Analytically evaluate the definite integral, resulting in a numerical value that represents the average electricity consumption for a full day.

Step-by-step solution

Define the Given Function

The power consumption as a function of time is given by \(C(t)=3.9-2.4 \sin (\pi t / 12)\). This function represents the power consumed by the household at different times during the day.

Compute the Integral

The total daily consumption can be calculated by integrating over the course of one full day, i.e., from \(t=0\) to \(t=24\). So, compute the definite integral \( \int_{0}^{24}C(t) dt = \int_{0}^{24} (3.9 - 2.4 \sin (\pi t / 12)) dt\).

Calculate the Antiderivative

Calculate the antiderivative resulting from the integration: \( C'(t) = 3.9t - [24*\cos(\pi t / 12) / \pi]\). Now evaluate the antiderivative at the limits of integration.

Evaluate at the Limits of Integration

Evaluate the antiderivative at \(t=24\) and \(t=0\), and subtract the two results: \( \Delta C = C'(24) - C'(0) \). This will provide the total electricity consumption over the course of the 24 hour day.

Interpret the Result

The result from the previous step represents the average daily power consumption in kilowatt-hours. This is the answer to the problem.