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Chapter 10: Problem 9

A certain appliance requires 225 watts when it is switched on. How much would it cost to run for \(m\) minutes, at a cost of \(d\) dollars per kilowatt-hour? Express your answer algebraically.

Short Answer

Expert verified
The cost to run the appliance for \(m\) minutes is \(C = 0.225 \times D \times (m/60)\) dollars.

Step by step solution

01

Convert power to kilowatts

Power given is 225 watts. Since cost is given per kilowatt-hour, convert this power to kilowatts. 1 kilowatt equals 1000 watts, so divide 225 by 1000 to get power in kilowatts. The equation become \(P = 225/1000 = 0.225 kW\)
02

Convert time to hours

Time is given in minutes. Since cost is given per kilowatt-hour, convert minutes to hours. 1 hour equals 60 minutes, so divide \(m\) by 60 to get time in hours. The equation becomes \(T = m/60 \) hours.
03

Calculate the cost

Now, multiply power by time and cost to find the total cost. The cost \(C\) for running the appliance for \(m\) minutes is given by the formula \( C = P \times D \times T\), where \(P\) is the power in kilowatts, \(D\) is the cost per kilowatt-hour and \(T\) is the time in hours. So, \(C = 0.225 \times D \times (m/60)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Conversion
When dealing with electrical appliances, the power they consume is often listed in watts. However, energy costs are usually expressed in terms of kilowatt-hours. Therefore, converting power from watts to kilowatts is essential.
To convert power from watts (W) to kilowatts (kW), you divide the number of watts by 1000, since 1 kilowatt equals 1000 watts.
For example, an appliance consuming 225 watts works as follows to convert to kilowatts:
\( P = \frac{225}{1000} = 0.225 \text{ kW} \)
This conversion is crucial because it aligns the power measurement with the units used for billing, making it easier to calculate energy costs accurately.
Time Conversion
Time conversion is an important step in energy cost calculations when determining how long an appliance has been running. This is because energy usage and cost are often calculated per hour.
If the time duration is provided in minutes, it should be converted into hours for consistency with kilowatt-hours.
One hour is equivalent to 60 minutes, so to convert minutes to hours, divide the number of minutes by 60.
For instance, if an appliance is running for \( m \) minutes, you convert it to hours using:
\( T = \frac{m}{60} \) hours.
This uniformity in time measurement ensures that energy computations are accurate and compatible with pricing structures.
Kilowatt-Hour Cost Calculation
Understanding the cost of running an appliance involves calculating how much energy it uses and the rate at which this energy is billed.
Once power and time are converted to kilowatts and hours respectively, you can calculate the actual cost.
The formula used is:
\( C = P \times D \times T \)
where \( C \) is the cost, \( P \) is the power in kilowatts, \( D \) is the cost per kilowatt-hour, and \( T \) is the time in hours.
By multiplying these values together, you determine the total cost to run the appliance for the given time period.
  • First, calculate the power in kW (which we've already done: 0.225 kW).
  • Second, compute the total time in hours.
  • Finally, multiply these by the cost per kilowatt-hour.
This arithmetic provides a clear and straightforward way to ascertain the energy expenses for any electrical device.

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