Problem 65 - Precalculus: Functions and Graphs

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 insulation cost.

Step 1: Calculate Monthly Savings

To determine how much money is saved each month due to insulation, we first calculate the reduction in heating costs. A 10% reduction on a monthly cost of \(200 is calculated as follows: \\[ \text{Savings per month} = 0.10 \times 200 = 20 \text{ dollars} \] This means the insulation saves \)20 each month.

Step 2: Determine Recovery Time in Months

To find out how many months it will take to recoup the installation cost, divide the total cost of insulation by the monthly savings. The formula is: \\[ \text{Number of months} = \frac{\text{Cost of insulation}}{\text{Monthly savings}} = \frac{2400}{20} = 120 \text{ months} \] Therefore, it takes 120 months to recover the initial insulation cost.

Key Concepts

Insulation Cost Analysis

When considering insulation, it's crucial to understand the initial expense involved. In our example, the installation cost for the insulation is \(\$2400\). This figure represents the upfront investment required to enhance the energy efficiency of the home.
An Insulation Cost Analysis involves evaluating the money spent on the insulation versus the benefits it brings over time. Here, the primary benefit is the reduction in heating expenses, specifically focusing on the long-term savings. Calculating this helps homeowners determine if the expense is justified.
This process also involves examining factors like the type of insulation, its effectiveness, local energy rates, and any possible rebates or incentives available. Keeping these factors in mind can help ensure the investment leads to substantial savings and energy conservation.

Monthly Savings Calculation

Calculating the monthly savings achieved through insulation is a simple yet vital step. In the given exercise, the insulation is expected to reduce heating costs by 10%.
To find out how much this reduction equates to per month:

• Take the average monthly heating cost, which is \(\\(200\).
• Calculate 10% of \(\\)200\), resulting in a savings of \(\\(20\) per month.

This calculation shows that every month, after the insulation is installed, the homeowner saves \(\\)20\) on heating. Such monthly savings, though small individually, add up over time and contribute significantly to energy efficiency. Being aware of these savings helps in planning the household budget more effectively.

Heating Cost Reduction

Reducing heating costs is a primary benefit of quality insulation. In the problem example, the insulation reduces heating expenses by 10%. This reduction isn't just about lower bills; it's also about improving energy usage efficiency.
Here's why heating cost reduction is important:

• It leads to consistent lower monthly bills, which accumulates to significant annual savings.
• Reduced heating use leads to less strain on heating systems, potentially extending their lifespan.
• Lower energy consumption contributes to environmental sustainability by reducing carbon footprints.

Understanding the way insulation impacts heating costs allows homeowners to better appreciate the value of investing in such home improvements. It's not only about recovering the initial costs but also about the long-term economic and environmental benefits.