Your revenue can be modeled by the function R(x) = 13x/x + 4. while your cost can be modeled by the function C(x) = x + 3.

Question

Determine a fully simplified rational expression for the profit where the profit made is the difference between revenue and cost

You are responsible for organizing fundraisers for different groups. A school calls and asks you to prepare a fundraiser for them. The school prefers that you keep things simple. You want to try something new, so you think about selling fancy mathematical calculators. You do some work, enter some data into the computer and come up with two equations that will help you.

Your revenue (how much money you bring in) can be modeled by the function R(x) = 13x/x + 4 while your cost (how much you need to pay out) can be modeled by the function C(x) = x + 3. The number of calculators you sell is given by x

a) Determine a fully simplified rational expression for the profit, P(x), where the profit made is determined by the difference between revenue and cost. State your restrictions!

b) Will the school be able to sell enough calculators to break even? Be sure to provide a complete solution, including the work and the justification for your answer.

c) If your model doesn’t result in a profit, determine how you can “fix” this problem. What can you do to ensure that your model produces a profit?

For this, you might get a negative x-value as an x-intercept. What does x represent? Does it make sense to have a negative x? This is a real world problem, not just some arbitrary math problem. you can't use a discriminant if you have some wild rational function, only for when a quadratic is equal to zero. However, you may be able to rearrange P(x)=0 to show that a quadratic equals zero

For this solution, you must provide the following:

 A diagram showing how the given information is used

 A full solution, showing all work

 Any justification/explanation that may be required

 A final statement that answers the given question

Answer 1

First I'm going to assume that the Revenue equation has a typo because 13x/x = 13, so I think the x in the denominator is supposed to be a number, let's say 2 in this case.

a) Like the prompt says you want to take P(x) = R(x) - C(x) = (13x/2+4) - (x+3) and simplify this, remember to distribute the negative. You can't sell less than 0 calculators, so x > 0 is your restriction. Also, x needs to be an integer (whole numbers) since you can't sell half a calculator.

b) To find the break even point, you need to set R(x)=C(x) and solve for x. Remember that x = calculators, so this will tell you how many calculators need to be sold to "break even". You can think about this in the real world. If you make (revenue) as much as you paid (cost), then you will break even. Notice that is the same as having 0 profit, or P(x) = 0 (and solve for x).

c) You can tell that you won't make a profit if your P(x) is negative for any x > 0 or like they are suggesting, the x-intercept of your profit function is a negative x value, but you can't sell a negative amount of calculators. To fix this, you would need more revenue, meaning that the number attached to the x value in the R(x) needs to be larger, which implies that you are charging more per calculator.

d) For the bold text, we know that x = calculators sold, so it makes no sense if x was negative, and is really concerning if your x-intercept of P(x) (which represents your break-even point) is negative.