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 may out y = f(x) and the graph of f(x) Is given below.
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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.
Note that pennies are allowed; for example it turns out that an input of $5.43 yields a payoff of $8.27. Of course the graph you are given does not have enough fine detail for you to obtain those exact numbers.
The problem is: how should you play to maximize your profit? What is your optimal input x?
After thinking about the problem for a while, you might well start to wonder what sort of access you have. Can you play only once? Can you play many times? How many times?
I will give you two options and you can choose whichever you wish.
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.
Which should you choose? In each case what is the optimal value of x and what is your total profit?
