Question

Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal \(=4184\) J. One hour of walking consumes roughly \(10^{6} \mathrm{J},\) or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where \(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s}\) Other useful units of power are kilowatts \(\left(1 \mathrm{kW}=10^{3} \mathrm{W}\right)\) and megawatts \(\left(1 \mathrm{MW}=10^{6} \mathrm{W}\right) .\) If energy is used at a rate of \(1 \mathrm{kW}\) for one hour, the total amount of energy used is 1 kilowatt-hour \(\left(1 \mathrm{kWh}=3.6 \times 10^{6} \mathrm{J}\right) .\) Suppose the cumulative energy used in a large building over a 24 -hr period is given by \(E(t)=100 t+4 t^{2}-\frac{t^{3}}{9} \mathrm{kWh},\) where \(t=0\) corresponds to midnight. a. Graph the energy function. b. The power is the rate of energy consumption; that is, \(P(t)=E^{\prime}(t) .\) Find the power over the interval \(0 \leq t \leq 24\) c. Graph the power function and interpret the graph. What are the units of power in this case?

Short answer

Question: Graph the function \(E(t) = 100t + 4t^2 - \frac{t^3}{9}\) representing the cumulative energy used in a building over a 24-hour period in kilowatt-hours. Then, find the power function P(t) and determine the power over the given interval [0, 24], graph P(t), and provide the units of power. Answer: The power function is \(P(t) = 100 + 8t - \frac{t^2}{3}\) and represents the power consumption over time in kilowatts (kW). The power at the beginning of the interval is 100 kW and decreases to 32 kW at the end of the 24-hour period.

Step-by-step solution

Graph the energy function E(t)

To graph the energy function, we can use any graphing tool like Desmos or Grapher. The given function is \(E(t) = 100t + 4t^2 - \frac{t^3}{9}\). Simply plug in the equation into the graphing tool to plot the energy function.

Find the power function P(t)

As the power is the rate of energy consumption, we need to find the derivative of the energy function. Given \(E(t) = 100t + 4t^2 - \frac{t^3}{9}\), differentiate E(t) with respect to t to get P(t): \(P(t) = \frac{dE(t)}{dt} = 100 + 8t - \frac{t^2}{3}\) Now, we have found the power function P(t).

Determine the power over given interval [0, 24]

Plug in the values given in the interval into the power function P(t): \(P(0) = 100 + 8(0) - \frac{(0)^2}{3} = 100 \thinspace kW\) \(P(24) = 100 + 8(24) - \frac{(24)^2}{3} = 32 \thinspace kW\) The power at the beginning of the interval is 100 kW, and at the end of the interval, it is 32 kW.

Graph the power function P(t) and interpret the graph

To graph the power function P(t), we can use the same graphing tool as we used to graph the energy function E(t). The function is \(P(t) = 100 + 8t - \frac{t^2}{3}\). Simply plug in the equation into the graphing tool to plot the power function. The graph will show the change in power consumption over time. It should start at 100 kW, decrease throughout the day and reach 32 kW at the end of the 24-hour period.

Determine the units of power

In this case, the energy function is given in kilowatt-hours (kWh), and since the power function is the derivative of the energy function, the units of power would be kilowatts (kW).

Key concepts

Energy Function

An energy function is a mathematical representation of how energy accumulates or is used over time. In this particular exercise, the energy function is defined as \( E(t) = 100t + 4t^2 - \frac{t^3}{9} \) kWh, where \( t \) represents time in hours starting from midnight.

This function allows us to calculate the total energy used by a building over a given period. To understand how the energy changes, it's helpful to plot this function on a graph. The shape and trajectory of the curve reveal how energy consumption accelerates or declines.

This step involves using tools like Desmos or Grapher to see a visual representation, which can make it easier to comprehend the variations in energy usage throughout the day. Studying the graph gives insights into patterns and trends in energy consumption.

Power Function

The power function is derived from the energy function and indicates the rate at which energy is consumed at any given time. Power, denoted \( P(t) \), is mathematically the derivative of the energy function \( E(t) \).

In differentiation, the derivative represents the rate of change. Thus, to find the power function from \( E(t) = 100t + 4t^2 - \frac{t^3}{9} \), we compute the derivative to obtain \( P(t) = 100 + 8t - \frac{t^2}{3} \).

This function describes how quickly or slowly energy is consumed in the building. It is dynamic, fluctuating throughout the 24-hour period as indicated by its specific values at different points in time. By evaluating \( P(t) \) at various intervals, we can better understand the changes in demand for energy, helping in planning and resource management.

Graphing Functions

Graphing functions is a crucial skill in visualizing mathematical concepts and relationships. When graphing the energy function \( E(t) \) and the power function \( P(t) \), the graphs depict energy usage and consumption rates over time.

These graphs not only help in understanding the behavior of functions but also in interpreting real-world situations, such as energy management in a building. For the energy function \( E(t) \), the graph demonstrates total energy usage as time progresses. On the other hand, the power graph shows variations in energy consumption rates indicated by \( P(t) \).

By using graphing tools, one can assign values to \( t \) and watch the graph evolve, giving a clear picture of when energy consumption peaks or drops within the day—useful information for optimizing energy use.

Derivatives in Calculus

Derivatives are foundational tools in calculus for determining how a function changes with respect to its variables. In this context, the derivative of the energy function \( E(t) \) with respect to time \( t \) provides us with the power function \( P(t) \).

Here are some vital points about derivatives:

• They measure the rate of change, showing how much a function is increasing or decreasing.
• In the case of energy functions, the derivative reflects how energy usage accelerates or decelerates over time.
• Understanding derivatives enables us to predict behavior, manage resources more effectively, and optimize processes in various applications beyond energy, such as economics or physics.

Mastering the use of derivatives and their applications is essential for making informed decisions based on changing conditions.