You're trying to decide whether to buy an energy-efficient \(225-\mathrm{W}\) refrigerator for \(\$ 1150\) or a standard \(425-\mathrm{W}\) model for \(\$ 850 .\) The standard model will run \(20 \%\) of the time, but better insulation means the energy-efficient model will run \(11 \%\) of the time. If electricity costs 9.5? \(\mathrm{kW} \cdot \mathrm{h}\), how long would you have to own the energy-efficient model to make up the difference in cost? Neglect interest you might earn on your money.

Understanding the electricity cost calculation is crucial when evaluating the efficiencies of household appliances. The fundamental formula to estimate the operating cost for an appliance over a given period is:
\[ \frac{(\text{Power in watts}) \times (\text{Percentage of time turned on}) \times (\text{Cost per kilowatt-hour})}{1000} \].
This equation helps in determining the cost of running an appliance per hour. As seen in the exercise, the power in watts is the appliance's consumption, the percentage represents its usage over the time, and the cost per kilowatt-hour reflects the current rate charged by the electricity provider. It's essential to convert watts to kilowatts by dividing by 1000 since electricity costs are typically expressed in cost per kilowatt-hour (\(\text{kWh}\)).
Practical Tip: Remember to align the units when performing these calculations. For instance, if the cost per kilowatt-hour is given in cents, ensure that all costs are calculated in cents to maintain consistency before converting to dollars if needed.

Break-even analysis is a powerful financial tool used to determine when an investment will start generating profit, compensating for its initial cost. In the context of energy-efficient appliances, this analysis helps identify the time it takes for the savings from the lower operational costs to cover the higher purchase price of the efficient model.
In our exercise, the break-even point is calculated by dividing the initial cost difference between the two models by the savings generated per hour of operation. This is demonstrated as:
\[ \frac{\text{Cost difference of the models}}{\text{Difference in operating costs per hour}} \].
It's essential to ensure that the cost difference and operating cost savings are in the same unit, typically cents, to perform accurate calculations. The result will provide you with the total hours needed for the energy-efficient appliance to become cost-effective compared to a less efficient alternative.

Comparing operational costs between different appliance models is a sensible approach to make an informed decision. By performing a thorough operational cost comparison, as illustrated in the provided solution, you can understand the long-term expenses associated with each option. It involves detailed analysis beyond just the initial purchase price.
Efficiency Matters: An energy-efficient model with a higher purchase price can result in lower operational costs over time, while a less expensive, less efficient model can have higher electricity usage costs. It is this comparison of hourly operational costs that uncovers potential savings—savings that can be beneficial in budgeting for households mindful of their energy consumption and long-term expenses. The key takeaway is to compare not just the prices at purchase, but the projected costs over the lifespan of the appliance.
Long-Term Perspective: When assessing appliances, factor in the longevity and expected operating times. An appliance's lifespan can significantly impact the total cost of ownership, making a seemingly costly energy-efficient model more economical in the long run.