Question

Solar panels provide \(65.0 \%\) of the heat for a certain building. If \(\$ 1560\) per year is now spent for heating oil, what would have been spent if the solar panels were not used?

Short answer

The total heating cost without the use of solar panels would have been \( \$1560 / 0.35 \approx \$4457.14 \) per year.

Step-by-step solution

Determine the percentage of heat provided by heating oil

Since solar panels provide 65.0% of the heat, the remaining heat, which is 35.0%, must be provided by heating oil. This is because the total percentage must add up to 100%.

Calculate the total heating cost with no solar panels

To calculate the total heating cost without the use of solar panels, divide the cost currently spent on heating oil by the percentage of heat it provides. Use the formula: Total cost = Cost of heating oil / Percentage of heating oil contribution. Remember to use the decimal form of the percentage.

Solve for the total heating cost

Substitute the given values into the formula: Total cost = \( \$1560 / 0.35 \). Perform the division to find the total heating cost.

Key concepts

Solar Panel Energy Savings

Harnessing the power of solar panels can lead to considerable energy savings for homeowners and businesses alike. The initial problem illustrates a scenario where 65% of a building's heating is provided by solar panels, which can translate into a significant reduction in dependence on traditional heating methods like oil or gas. When a building integrates solar panel systems to its infrastructure, not only does it reduce the carbon footprint, but it also ensures a decrease in the recurring costs of energy.

Understanding these savings starts with grasping what percentage of the energy comes from renewable sources, in this case, solar panels. The percentage is crucial because it defines the fraction of total heating costs that can be eliminated due to solar panels. To calculate the potential savings, one must consider the cost that would be incurred if no solar panels were used, which really puts into perspective the financial impact of choosing sustainable energy sources.

Calculating Heating Costs

When facing a cold climate or winter season, heating costs can become a significant portion of a household's expenses. The exercise implies a need to understand how much would be spent solely on heating oil if solar panels weren't used. Initially, the exercise provides that heating oil only covers 35% of the total heating due to the presence of solar panels.

Calculation of heating costs without any contribution from solar panels requires an understanding of percentages and their conversion into monetary values. Using the information that $1560 covers 35% of the heating needs, the total cost can be extrapolated. It's crucial at this point to convert percentage to a decimal by dividing by 100, a key step many students overlook. By dividing the current heating oil cost by the decimal form of the percentage it represents, one can determine the total heating costs that would have been incurred if the building relied solely on heating oil.

Mathematical Problem-Solving

Mathematical problem-solving involves a systematic approach to understanding and solving mathematical problems. This exercise takes students through a real-world application of percentages, dividing one quantity by another, and comprehending the impact of their findings in a practical context.

In problem-solving, it is vital to first identify what is known and what is to be found – in our case, the percentages of heat provided by two sources and the total cost without one of them. The next step involves choosing the correct mathematical operations and formulas to use. Here, by simply using the formula Total cost = Cost of heating oil / Percentage of heating oil contribution and substituting the given numbers, we solve for the unknown. Such exercises enhance a student's ability to apply mathematics to everyday situations, improving both their numerical skills and their understanding of how mathematics can be a tool for decision-making in areas such as financial planning and energy management.