Question

Aidan buys used bicycles, fixes them up, and sells them. His average cost to buy and fix each bicycle is $47. He also incurred a one-time cost of

$840 to purchase tools and a small shed to use as his workshop. He sells bikes for $75 each. Use this information for the exercises.


WRITE Write revenue and cost functions R(x) and C(x) for Aidan's situation, where x is the number of bicycles. How do you include the one-time
cost in C(x)?

WRITE Write a profit function P(x) such that P(x) = R(x)-C(x). In words, what does P(x) represent?


PERSEVERE List key features for the profit function P(x). Then use the key features to sketch a graph, on a separate sheet of paper, that shows
the profit P(x) as a function of x bicycles.


ANALYZE Which key feature of the graph represents Aidan's break-even point (profit = 0)? Explain how to use your graph to find the most
accurate value for this feature.

Answer 1

P(x) = $31x - $840

Explanation:

Let x be the number of bicycles. The total cost for purchasing and repairing the bikes is $47x, plus a one-time purchase of $840 for a workshop and tools. Total cost, C is therefore:

C(x) = $47x + $840

We learn Aidan sells the bikes for $75 each. Total revenue, R, is:

R(x) = $75x

Profit, P, would be the difference of revenue, R, and costs, C:

P(x) = R(x) - C(x) or $75x -($47x + $840)

P(x) = $75x -$47x - $840

P(x) = $31x - $840

P(x) represents the total profit for x bicycles.

See the attached graph. Note the key features of the graph are the breakeven (27 bikes) and the initial investment (-$840). We can find the breakeven point by looking for the value of x when the profits are $0. A more accurate determination can be found by solving the equation we developed for the case in which P(x) = $0:

P(x) = $31x - $840

0 = $31x - $840 for P(x) = 0 (breakeven)

$31x = $840

x = 27.1 units. We need to round to 27 units since 0.1 of a bicycle makes no sense in this context.