The Gordon family plans to buy a TV. One TV has a purchase price of $330 and an estimated yearly operating cost of $14. The other has a purchase price of $369 and an estimated yearly operating cost of $9. Which TV should the Gordons buy if they plan to keep it for 8 years? Use rational functions to help justify your answer.

Answer:

To determine which TV is a better value over 8 years, we can use rational functions to model the average annual cost. For the first TV, the average annual cost is f(x) = (330 + 14x) / x. For the second TV, the average annual cost is g(x) = (369 + 9x) / x. Evaluating both functions for x = 8 years: f(8) = (330 + 14 * 8) / 8 = 442 / 8 = $55.25 g(8) = (369 + 9 * 8) / 8 = 441 / 8 = $55.125 (or approximately $55.13) Since the average annual cost (and therefore the total cost) for the $369 TV is lower after 8 years, the Gordon family should buy the $369 television.