Cost, Revenue, and Profit
The cost function (C) usually comes in two parts: A fixed cost and a variable cost.
The revenue (or income R) is the unit price of the service times the amount of service provided.
The profit function (P) is the difference between these two:
P = R - C
The optimum operating point of the business is where the profit is positive and maximum.
The fixed cost for P&F is:
$65,000 + $1,000 = $66,000
The variable daily cost is:
$1.x per car
Where x is the number of cars it gives service to. If P&F operates 30 days a month, then the variable monthly cost is:
$30.x
The total monthly cost for P&F is:
C = $66,000 + $30x
The revenue comes from taking cars in and out of the airport for which it charges $8.50 per day per car. If x cars are served each day for a month, then the total revenue is:
R = $8.50 * 30 * x
R = $255x
a. The break-even point occurs when the cost and the revenue are equal, thus:
$255x = $66,000 + $30x
Subtracting 30x:
225x = 66,000
Dividing by 225:
x = 293.3
Thus, P&F needs to service 294 cars per day to break-even
b. Since the revenue is a linear function of the price, there is not a maximum revenue when varying the price. Revenue is higher if the price is higher at every moment. For example, if it charges $10 per car per day, the revenue would be:
R = $10 * 30 * x =$300x
c. The profit function is:
P = $255x - ($66,000 + $30x)
P = 225x - 66,000
This function is also linear with the price. The higher the price, the higher the profit. For example, charging $10 per car per day:
P = 270x - 66,000
And the break-even point will occur at x = 66,000/ 270 = 245 cars per day.
d. To obtain $60,000 in profit:
225x - 66,000 = 60,000
Adding 66,000:
225x = 66,000 + 60,000
225x = 126,000
x = 560
P&F should make 560 sales to make $60,000 in profit