Problem 233 - Calculus Volume 1

Question

The following table lists the electrical power in gigawatts - the rate at which energy is consumed- used in a certain city for different hours of the day, in a typical 24 -hour period, with hour 1 corresponding to midnight to 1 a.m. Find the total amount of power in gigawatt-hours (gW-h) consumed by the city in a typical 24 -hour period.

Step-by-Step Solution

Answer

Sum the gigawatt values for each hour to find total gW-h.

Step 1: Understand the Unit Measurement

Gigawatts (gW) measure the rate of energy consumption, while gigawatt-hours (gW-h) measure the total energy consumed over time. To convert gigawatts to gigawatt-hours over the span of 1 hour, multiply the power (in gW) by time (in hours). Since each hour consumes power at the rate specified, the power value for each hour directly translates to gigawatt-hours for that hour because it is a 1-hour period.

Step 2: List Power Consumption for Each Hour

Note the gigawatt value for each of the 24 hours. For example, if hour 1 (midnight to 1 a.m.) has a consumption of 3 gW, then this hour contributes 3 gW-h to the total energy consumption.

Step 3: Calculate Total Power Consumption

Sum up the power values for each hour. This yields the total energy consumption over the 24-hour period because each hour has a consumption equivalent to its rate due to the 1-hour duration of each period.

Step 4: Interpret the Result

The sum you calculated represents the total gigawatt-hours consumed over the entire day, which is the total energy the city uses in a typical 24-hour period.

Key Concepts

Unit Conversion

When calculating energy consumption, it's important to understand the difference between various units of measurement. In this exercise, we deal with gigawatts (gW) and gigawatt-hours (gW-h). Gigawatts measure the rate at which energy is used, while gigawatt-hours measure the total amount of energy consumed. Think of gigawatts as the speed of energy consumption and gigawatt-hours as the total distance covered over time.
In simpler terms, to convert from a power rate (gW) to a total energy consumption (gW-h) over a period, you multiply the power by the number of hours. For example, 1 gigawatt over 1 hour equals 1 gigawatt-hour (1 gW × 1 h = 1 gW-h). This is because each hour's consumption rate directly translates to an energy amount for that hour.

Power Consumption

Power consumption refers to the amount of energy used over a given period. In the context of our exercise, this is about understanding how much power a city consumes each hour. It's measured in gigawatts, which indicates the energy rate. By examining each hourly energy usage, typically laid out in a schedule, we can construct a understanding of daily consumption.

• For every hour, note the corresponding power consumption. This is typically provided in gigawatts.
• Each hourly power rate is used for precisely one hour, meaning the gigawatts can be directly translated to gigawatt-hours for that hour alone with a 1:1 conversion.
• Summing up these values will give a comprehensive view of how much energy the city uses across the day.

Analyzing power consumption trends is crucial for city planners to optimize energy use and create sustainable strategies.

Total Energy Calculation

The total energy calculation is the culmination of converting each hourly energy consumption into gigawatt-hours and summing them across the entire day. This gives the total energy consumed by the city.

Calculating Total Energy

To find the total energy the city uses within a 24-hour period, perform the following:

• List out the energy consumption rates provided for each hour in gigawatts.
• Since they represent the power used over exactly 1 hour, these values also indicate gigawatt-hours.
• Add all these gigawatt-hour values together.

The result is the total gigawatt-hours, which tells you how much energy was used in that one day period. This kind of calculation is vital for understanding energy needs and planning for efficient energy distribution and consumption.