Problem 54 - Technical Mathematics with Calculus

Question

A house that costs \(\$ 635\) per year to heat has insulation installed in the attic, causing the fuel bill to drop to \(\$ 518\) per year. Find the percent change in fuel cost.

Step-by-Step Solution

Answer

The percent change in fuel cost is 18.43%

Step 1: Calculate the Change in Cost

Find the difference between the original fuel cost and the new fuel cost. Subtract the new fuel cost from the original fuel cost: \(635 - \)518.

Step 2: Calculate the Percent Change

Take the change in cost and divide it by the original cost. Then, to find the percentage, multiply the result by 100.

Step 3: Write Down the Percent Change Formula

The formula for percent change is Percent Change = (Change in Value / Original Value) * 100%.

Step 4: Plug in the Values and Solve

Substitute the values from Step 1 into the formula: Percent Change = ((\(635 - \)518) / $635) * 100%.

Key Concepts

Percentage Decrease

Understanding the concept of percentage decrease is essential in dealing with daily financial matters, like the fuel cost savings in our exercise. A percentage decrease occurs when the new value of an item is less than its original value.
To find the percentage decrease, one subtracts the new value from the original value, divides the difference by the original value, and then multiplies the result by 100 to convert it to a percentage.
For example, if a product costs \(100 and then is reduced to \)80, the price has decreased. To determine the percentage decrease:

• Calculate the difference: \(100 - \)80 = \(20.
• Find the percentage: (\)20 / $100) \times 100 = 20%.

So, the product is now 20% cheaper than before. This calculation is handy for budgeting and comparing costs over time.

Fuel Cost Savings

The concept of fuel cost savings comes into play when we evaluate the financial benefits of energy-efficient improvements. In our exercise, the installation of insulation results in a reduction in the heating fuel bill.
To assess the savings made, one must look at the difference in cost before and after the energy-saving measure was implemented. This is not just a simple subtraction; the result demonstrates the financial impact of the decision to insulate.
These savings can be a critical factor in personal finance or when managing the budget of a business or organization. Calculating fuel cost savings also encourages the adoption of eco-friendly practices, as they significantly contribute to reduced expenditures over time.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that goes beyond classroom learning. It involves recognizing patterns, understanding formulas, and applying logic to find solutions to complex issues.
In our insulation example, several steps are involved in the problem solving process:

• Determining the required information.
• Identifying the correct formula to utilize.
• Performing the necessary calculations.
• Interpreting the results within the context of the problem.

This structured approach helps break down complex problems into manageable parts, ensuring each step is logical and advances one towards the final answer. It's a process that's valuable not just for academic exercises but for tackling real-world scenarios that require analytical thinking and decision-making.