The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a geothermal heating system. The homeowner estimates that the average annual energy cost will then be $2,712. Which of the following inequalities can be solved to find t, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?

Options

A. 25,000 gt (4,334 − 2,712) t
B. 25,000 lt (4,334 − 2,712) t
C. 25,000 − 4,334 gt 2,712t
D. `25,000 gt (4,332)/(2,712)t`

Answer

B

Explanation

The savings each year from installing the geothermal heating system will be the average annual energy cost for the home before the geothermal heating system installation minus the average annual energy cost after the geothermal heating system installation, which is (4,334 − 2,712) dollars. In t years, the savings will be (4,334 − 2,712)t dollars. Therefore, the inequality that can be solved to find the number of years after installation at which the total amount of energy cost savings will exceed (be greater than) the installation cost, $25,000, is 25,000 lt (4,334 − 2,712)t.
Choice A is incorrect. It gives the number of years after installation at which the total amount of energy cost savings will be less than the installation cost. Choice C is incorrect and may result from subtracting the average annual energy cost for the home from the onetime cost of the geothermal heating system installation. To find the predicted total savings, the predicted average cost should be subtracted from the average annual energy cost before the installation, and the result should be multiplied by the number of years, t. Choice D is incorrect and may result from misunderstanding the context. The ratio `(4,332)/(2,712)` compares the average energy cost before installation and the average energy cost after installation, it does not represent the savings.