A company that produces custom

Question

A company that produces custom license plates spends $5525 per month in building overhead plus $2.50 per plate. If each plate sells for $5.99how many plates must the company sell before making a profit?

Answer 1

In problems like this, it helps to reframe the information in terms of what the company earns (the revenue) and what the company spends (the cost).  In this case, the revenue is just 5.99 per license plate sold, so the total revenue is 5.99x, where x is the number of license plates it sold.  The cost has a fixed amount of 5525 per month no matter what, plus an additional 2.50 per license plate, meaning you have a total cost of 5525+2.5x.

 

In order for the company to make a profit, the revenue must be greater than the cost.  If we think about this in terms of linear equations, we need the revenue function y=5.99x to be greater than the cost function y=2.5x+5525.  We know it will be eventually, because it has the bigger slope and so it's a steeper line, but the cost function has a higher y-intercept (5525 compared to 0), so it starts off a lot higher.  In order to figure out where revenue is greater than cost, we just have to solve the system of the two equations.  Substitution is easiest with these two (they're both already written in terms of y):

 

5.99x=2.5x+5525

 

Subtract 2.5x from both sides to get

 

3.49x=5525

 

Divide both sides by 3.49 to get

 

x=1583.09 (rounded to the nearest hundredth).

 

So, at that x-value, the revenue is equal to the cost.  In order for the company to make a profit, it must sell 1584 license plates per month.