Money Machine

You have access to a magical money machine.  You can put in any amount of money you want, between 0 and $10, and pull the big brass handle, and some “payoff” will come pouring out.  Now this payoff depends on the amount you put in.  If you feed it x dollars it will pay out , and the graph of  is given at the right. 

For example, an input of $2 will yield a payoff of $4, whereas an input of $5 will yield a payoff of $8. Just to be clear, if you put in $5 and get $8 back, your profit or net gain is 8 – 5 = $3. 

The problem is: how should you play to maximize your profit?  What is your optimal input x?

Well the answer depend on what kind of access you have to the machine and here we consider two options.

• Option A.  You are allowed to pull the handle a total of 100 times.

• Option B.  The total amount you feed into the machine cannot exceed $300.

We will look at both a graphical and an algebraic solution.

This is an excellent grade 11 problem. It develops strategic thinking and it fosters the connection between algebraic and graphical analysis.