A small company heats its building using a gas furnace. It expects to spend ​$9 comma 072 on natural gas this first year. The cost of the natural gas is expected to increase by 8​% per year every subsequent year. There is also the maintenance cost on the gas furnace which is expected to be ​$851.00 in the first year and to increase by 15​% per year every subsequent year. The planning horizon is 14 years and the company uses an annual interest rate of 18​% for economic analysis of its investments.
Question content area bottom
Part 1
​(a) The projected annual cost of natural gas in year 14 is ​$24,667 ​
​(b) The projected annual maintenance cost of the furnace in year 14 is ​$
  

We are asked to calculate the projected annual cost of natural gas and maintenance in year 14 for a small company. Here's how we approach both parts:

### Part (a): Projected Annual Cost of Natural Gas in Year 14
We know the following:
- The initial cost of natural gas is $ 9072 $ in the first year.
- The cost increases by $ 8\% $ every year.
- We are calculating the cost in year 14.

The formula for the cost of gas after $n$ years with an annual increase is:

\[
C_{\text{gas},n} = C_{\text{gas},0} \times (1 + r)^{n-1}
\]

where:
- $C_{\text{gas},n}$ is the cost in year $n$,
- $C_{\text{gas},0} = 9072$ is the initial cost of gas,
- $r = 0.08$ (the rate of increase),
- $n = 14$ is the year.

So the cost in year 14 is:

\[
C_{\text{gas},14} = 9072 \times (1 + 0.08)^{14-1} = 9072 \times (1.08)^{13}
\]

### Part (b): Projected Annual Maintenance Cost in Year 14
We are given:
- The initial maintenance cost is $ 851 $ in the first year.
- The maintenance cost increases by $ 15\% $ every year.
- We need to calculate the maintenance cost in year 14.

Using the same growth formula:

\[
C_{\text{maint},n} = C_{\text{maint},0} \times (1 + r)^{n-1}
\]

where:
- $C_{\text{maint},n}$ is the maintenance cost in year $n$,
- $C_{\text{maint},0} = 851$ is the initial maintenance cost,
- $r = 0.15$ (the rate of increase),
- $n = 14$ is the year.

Now let's compute both the projected gas cost and maintenance cost for year 14.The projected costs in year 14 are as follows:

- **Projected annual cost of natural gas**: \$24,672.43
- **Projected annual maintenance cost**: \$5,236.02

These projections account for the annual increases of 8% for natural gas and 15% for maintenance costs.

Do you want further details on how these calculations were performed or have any other questions?

Here are some related questions:
1. How can we calculate the total cost over the 14 years, including both gas and maintenance?
2. What would be the impact if the gas price increase rate changed to 10% instead of 8%?
3. How does the interest rate of 18% influence long-term cost predictions?
4. Can we compute the net present value (NPV) of the total cost over 14 years?
5. How do different inflation rates affect the overall cost prediction?

**Tip**: Always use exponential growth formulas for projecting costs with constant percentage increases over time.

A small company heats its building using a gas furnace. It expects to spend $9,072 on natural gas this first year. The cost of the natural gas is expected to increase by 8% per year every subsequent year. There is also the maintenance cost on the gas furnace which is expected to be $851.00 in the first year and to increase by 15% per year every subsequent year. The planning horizon is 14 years and the company uses an annual interest rate of 18% for economic analysis of its investments.

Calculate the projected annual cost of natural gas and maintenance over 14 years for a company using a gas furnace. This problem involves exponential growth calculations with 8% annual increase in gas costs and 15% annual increase in maintenance costs. Suitable for college-level economic and mathematical analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation