Question

Consider a homeowner who is replacing his 25 -year old natural gas furnace that has an efficiency of 55 percent. The homeowner is considering a conventional furnace that has an efficiency of 82 percent and costs \(\$ 1600\) and a high efficiency furnace that has an efficiency of 95 percent and costs \(\$ 2700 .\) The homeowner would like to buy the high-efficiency furnace if the savings from the natural gas pay for the additional cost in less than 8 years. If the homeowner presently pays \(\$ 1200\) a year for heating, determine if he should buy the conventional or high-efficiency model.

Short answer

Answer: ___________

Step-by-step solution

Calculate annual gas cost with the old furnace

To find the annual gas cost with the old furnace, we will use the formula: Annual gas cost = Heating cost / Old furnace efficiency Where: Heating cost = $1200 (provided in the problem) Old furnace efficiency = 55% So the annual gas cost will be: Annual gas cost = $1200 / 0.55

Calculate annual gas cost with the conventional and high-efficiency furnaces

Using the same formula as in step 1, we'll find the annual gas cost for both furnaces as follows: For conventional furnace (82% efficiency): Annual gas cost = Heating cost / Conventional furnace efficiency Annual gas cost = $1200 / 0.82 For high-efficiency furnace (95% efficiency): Annual gas cost = Heating cost / High-efficiency furnace efficiency Annual gas cost = $1200 / 0.95

Calculate annual gas savings

To find the annual gas savings, we need to subtract the cost of the new furnaces from that of the old furnace: Conventional furnace gas savings = Old furnace gas cost – Conventional furnace gas cost High-efficiency furnace gas savings = Old furnace gas cost – High-efficiency furnace gas cost

Calculate the payback period

The payback period is the time it takes for the savings of the high-efficiency furnace to cover its additional cost. We'll use the following formula: Payback Period = (High-efficiency furnace cost - Conventional furnace cost) / (High-efficiency furnace gas savings - Conventional furnace gas savings) Where: High-efficiency furnace cost = $2700 Conventional furnace cost = $1600 By calculating the payback period, we can determine if the high-efficiency furnace is worth the extra cost.

Compare the payback period to 8 years

If the payback period calculated in step 4 is less than 8 years, then the homeowner should buy the high-efficiency furnace, otherwise, they should buy the conventional furnace. Compare the calculated payback period to 8 years and make the appropriate recommendation based on the result.