Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

The cost of installing insulation in a particular two-bedroom home is \(\$ 2400\). Present monthly heating costs average \(\$ 200\), but the insulation is expected to reduce heating costs by \(10 \%\). How many months will it take to recover the cost of the insulation?

Step-by-Step Solution

Answer

It will take 120 months to recover the cost of insulation.

Step 1: Identify the Cost Savings per Month

First, determine the amount saved per month on heating costs by installing the insulation. The current heating cost is \( \\(200 \), and the insulation saves \( 10\% \) of the cost. Calculate \( 10\% \) of \( \\)200 \) as follows: \( \\(200 \times 0.10 = \\)20 \). Therefore, the monthly saving after insulation is \( \$20 \).

Step 2: Calculate Payback Period

To find out how many months it will take to recover the cost of insulation, divide the total insulation cost by the monthly savings. The insulation cost is \( \\(2400 \) and the monthly savings are \( \\)20 \). Thus, the number of months required is calculated as: \( \frac{\\(2400}{\\)20} = 120 \).

Key Concepts

Insulation Cost

When considering home improvements, such as adding insulation, it's essential to understand the initial costs involved. Insulation cost refers to the expense incurred to install insulation materials and labor in a home. In our example, the insulation cost is set at $2400. This cost needs to be thought of as an investment, aimed at saving money on future utility bills by boosting energy efficiency. Before proceeding with such improvements, it's a good idea to compare different insulation options and brands to ensure you receive value for your money. Look into factors such as material type, thickness, and the insulating properties often described by the R-value. The higher the R-value, the better the insulation. Hence, understanding insulation cost involves knowing how much you need to spend and what you can expect as a return.

Heating Cost Savings

Heating cost savings are the reductions in energy expenses resulting from improved energy efficiency, such as through installing insulation. With proper insulation installed, less heat escapes from your home, thereby reducing the amount of energy required to maintain comfortable indoor temperatures.In our example, the monthly heating cost without insulation is \(200. Once the insulation is added, a 10% reduction in this cost is expected. The savings calculation is simple:

• Monthly savings: 200 \times 0.10 = 20
• Thus, each month the homeowner saves \)20 on heating.

These savings gradually accumulate, helping offset the initial cost of insulation. It's vital to monitor these savings over time to understand when the investment effectively starts to pay off.

Payback Period

The payback period of an investment like home insulation indicates how long it will take for the investment to "pay for itself" through cost savings. It is crucial for financial planning and helps in understanding the efficiency of the investment in energy savings.To calculate the payback period, you would divide the total investment cost by the monthly cost savings:\[\text{Payback Period} = \frac{\text{Total Insulation Cost}}{\text{Monthly Savings}} = \frac{2400}{20} = 120 \text{ months}\]This shows it will take 120 months, or 10 years, for the savings to cover the initial insulation cost. Knowing the payback period can greatly influence a homeowner's decision to proceed with installing insulation, as they can weigh the long-term benefits against the upfront expense.

Percentage Savings

Percentage savings refer to the proportion of cost savings expressed as a percentage of the total cost. In the context of home insulation, it shows how much of the heating costs are reduced due to the insulation. For this scenario, the formula to determine the savings percentage is applied to the monthly heating cost:

• Original cost: $200 per month.
• Savings: 10% of $200 is $20.
• This means that 10% of the monthly heating cost is being saved each month.

This percentage is crucial when analyzing energy efficiency solutions, as it provides a clear picture of how impactful the solution is. A higher percentage denotes more significant savings, which can be a persuasive factor when considering such home improvements.