Problem 42 - Calculus Single Variable

Question

A kilowatt-hour is the amount of energy consumed in 1 hour at the constant rate of 1000 watts. At time \(t=0\) hour, a three-way lamp is turned on at the 50 -watt setting. An hour later, the lamp is turned up to 100 watts. Forty minutes after that, the lamp is turned up to 150 watts. Ninety minutes later, it is turned off. Let \(E(t)\) be the (cumulative) energy consumption in kilowatt-hours expressed as a function of time measured in hours. Graph \(E(t)\) for \(0 \leq t \leq 4\)

Step-by-Step Solution

Answer

E(t) increases at specific intervals, stabilizing at 0.3417 kWh post 3.17 hours.

Step 1: Understand the Problem

We want to express the energy consumption of a lamp as a function of time in kilowatt-hours (kWh). Given its operations: 50 watts for 1 hour, 100 watts for 40 minutes, and 150 watts for 90 minutes, we need to calculate the energy consumed at each stage and sum them.

Step 2: Convert Watts to Kilowatts

Energy consumption should be in kilowatt-hours. Since 1 kilowatt (kW) is 1000 watts (W), we first convert the power usage from watts to kilowatts: 50 W = 0.05 kW, 100 W = 0.10 kW, and 150 W = 0.15 kW.

Step 3: Calculate Energy Consumption in Each Time Interval

For each time interval, multiply the power usage in kilowatts by the time period in hours. - First hour (0 to 1): 0.05 kW * 1 h = 0.05 kWh. - Next 40 minutes (1 to 1.67 hours since 40 minutes is 2/3 of an hour): 0.10 kW * (2/3) h = 0.0667 kWh. - Final 90 minutes (1.67 to 3.17 hours since 90 minutes is 1.5 hours): 0.15 kW * 1.5 h = 0.225 kWh.

Step 4: Calculate Cumulative Energy Consumption

Add up the energy consumed in each time interval to find the cumulative energy consumption as a function of time: - At 1 hour: E(1) = 0.05 kWh. - At 1.67 hours: E(1.67) = 0.05 + 0.0667 = 0.1167 kWh. - At 3.17 hours: E(3.17) = 0.1167 + 0.225 = 0.3417 kWh.

Step 5: Graph the Function

Graph each segment of the function describing energy consumption: - From t=0 to t=1, E(t)= 0.05 kWh is constant. - From t=1 to t=1.67, with a slope of 0.0667, showing linear increase. - From t=1.67 to t=3.17, with a slope of 0.225, showing another linear increase.

Step 6: Draw Energy Function Conclusion

The resulting graph consists of three segments, starting with a constant value, followed by two linear increases. E(t) stabilizes at 0.3417 kWh after t=3.17 hours. The function E(t) reflects the cumulative energy used as the lamp stays on and transitions between power settings.

Key Concepts

Kilowatt-hour

A kilowatt-hour (kWh) is a unit of energy that quantifies the amount of electrical energy used over time. It represents the usage of 1000 watts of power continuously for one hour. Understanding kWh is fundamental when calculating energy consumption for devices in household and commercial settings.

• Power Conversion: Kilowatts (kW) is the standard unit of power, where 1 kW equals 1000 watts (W).
• Time Factor: The unit hour indicates the duration over which this power is utilized.
• Calculation: To calculate energy consumption in kWh, multiply the power of the appliance (in kilowatts) by the time it operates (in hours).

In our exercise, the lamp's energy consumption is calculated by understanding how it transitions between different wattage levels over specific time intervals, ultimately transitioning into kilowatt-hours for interpretation.

Time Interval

Time interval is the duration for which a certain state or condition persists. It is essential when calculating energy consumption, as energy is often assessed over specific periods. In the exercise, the lamp operates at different wattages during three distinct time intervals.

• Identifying Intervals: The exercise involves 1 hour at 50 watts, 40 minutes (which is two-thirds of an hour) at 100 watts, and 90 minutes (1.5 hours) at 150 watts.
• Conversion: Breaking down minutes into hours (e.g., 40 minutes is \(\frac{2}{3}\) hour and 90 minutes is 1.5 hours), facilitates precise calculations.

The careful consideration of these time intervals is crucial. Accurately assessing these durations ensures that calculations of energy consumption yield correct results and provide a clear picture of energy usage over time.

Cumulative Function

A cumulative function combines values over time, displaying an accumulated total. In the context of energy consumption, it sequentially adds the energy used over consecutive time intervals, creating a complete picture of total energy usage.

• Sequential Addition: Begin with the energy consumed in the first interval and add the amounts from each subsequent interval.
• Real-world Application: For the lamp, it starts with 0.05 kWh after the first hour, adds 0.0667 kWh after the next interval, and concludes with a total of 0.3417 kWh by the end of the third interval.

By creating such a function, you not only assess energy usage at any time within the period of interest but also understand overall energy trends. This cumulative approach is vital for seeing the bigger energy consumption picture and allows for more informed decision-making.

Graphing Functions

Graphing functions visually represents data or mathematical expressions, providing an intuitive understanding of how variables change over time. In the case of the lamp's energy consumption, the graph forms three distinct segments depicting constant and increasing usage rates.

• Segments: The graph starts with a constant value, symbolizing a steady rate at 50 watts for the initial hour, followed by linear increases representing changing wattage levels.
• Slope Representation: Each segment's slope reflects the rate of energy consumption over its interval—from flat (constant) to upwards (increasing).
• End Process: After 3.17 hours, the graph stabilizes, showing that no more energy is consumed post this duration as the lamp is turned off.

The graphical analysis of energy functions not only demonstrates changes over time but also helps in identifying patterns or anomalies in energy usage. It effectively communicates information that may be less clear via numerical data alone, making it an invaluable tool in understanding energy dynamics.