A small company heats its building using a gas furnace. It expects to spend 

​$8 comma 8008,800 

on natural gas this first year. The cost of the natural gas is expected to increase by 

1010​% 

per year every subsequent year. There is also the maintenance cost on the gas furnace which is expected to be 

​$874.00874.00 

in the first year and to increase by 

1515​% 

per year every subsequent year. The planning horizon is 

1515 

years and the company uses an annual interest rate of 

18​% 

for economic analysis of its investments. (a) The projected annual cost of natural gas in year 

15

,(b) The projected annual maintenance cost of the furnace in 15 year , (c) The total present worth of operating and maintaining the furnace over the 

15

years 

To solve this problem, we'll approach it step-by-step, breaking it down into three parts:

### Part (a) – Projected Annual Cost of Natural Gas in Year 15
The cost of natural gas increases by 10% per year. This means the cost of gas in year $ n $ follows the formula of compound interest:

\[
C_n = C_1 (1 + r_{\text{gas}})^{n-1}
\]

Where:
- $ C_1 = 8,800 $ (the cost in the first year)
- $ r_{\text{gas}} = 0.10 $ (the annual increase rate for natural gas)
- $ n = 15 $ (the year we want to calculate for)

We can plug the values into this formula to find the cost of natural gas in year 15.

### Part (b) – Projected Annual Maintenance Cost of the Furnace in Year 15
The maintenance cost increases by 15% per year, so we can apply a similar formula as in part (a) for the maintenance cost in year $ n $:

\[
M_n = M_1 (1 + r_{\text{maint}})^{n-1}
\]

Where:
- $ M_1 = 874 $ (the cost of maintenance in the first year)
- $ r_{\text{maint}} = 0.15 $ (the annual increase rate for maintenance)
- $ n = 15 $

We can plug in the values to calculate the maintenance cost in year 15.

### Part (c) – Total Present Worth of Operating and Maintaining the Furnace Over 15 Years
The total present worth (PW) is calculated by discounting the future costs back to present value. For a future amount $ C_n $ or $ M_n $ occurring in year $ n $, the present value is:

\[
PV = \frac{F}{(1 + i)^n}
\]

Where:
- $ F $ is the future cost in year $ n $
- $ i = 0.18 $ (the annual interest rate)
- $ n $ is the year in which the cost occurs.

The total present worth of operating and maintaining the furnace over 15 years will be the sum of the present worths of all future gas and maintenance costs. We'll need to sum up each year's cost over 15 years using this formula.

Let me first calculate the projected costs for year 15 and then proceed to calculate the total present worth.### Results:
(a) The projected annual cost of natural gas in year 15 is **$33,417.99**.

(b) The projected annual maintenance cost of the furnace in year 15 is **$6,184.17**.

(c) The total present worth of operating and maintaining the furnace over 15 years is **$80,959.65**.

Would you like any details on the calculations or further explanations?

### Related Questions:
1. How would a different annual interest rate impact the present worth?
2. What if the gas price increase was 5% instead of 10%? How would that change the future costs?
3. How do we compute present worth for irregular payment schedules?
4. How would using a discount rate of 20% affect the present worth calculation?
5. What is the difference between simple and compound interest in this type of economic analysis?

### Tip:
When comparing long-term investments, always factor in the discount rate to account for the time value of money.

A small company heats its building using a gas furnace. It expects to spend $8,800 on natural gas this first year. The cost of the natural gas is expected to increase by 10% per year every subsequent year. There is also the maintenance cost on the gas furnace which is expected to be $874.00 in the first year and to increase by 15% per year every subsequent year. The planning horizon is 15 years and the company uses an annual interest rate of 18% for economic analysis of its investments. (a) The projected annual cost of natural gas in year 15, (b) The projected annual maintenance cost of the furnace in year 15, (c) The total present worth of operating and maintaining the furnace over the 15 years.

Explore how to calculate the projected annual costs of natural gas and maintenance for a gas furnace over 15 years, using compound interest formulas and economic analysis. The problem involves calculating future gas and maintenance costs, and determining the present worth over 15 years with an 18% interest rate.

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