4.3 The money-making machine.

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1 . The money-making machine. You have access to a magical money making machine. You can put in any amount of money you want, between and $, and pull the big brass handle, and some payoff will come pouring out. Now this payoff is found to be somewhat unpredictable; that is, the same amount, put in a second time, will typically yield a different payoff. However, what is known is the average payoff A = A() for each input, and this is graphed below. output A payoff graph input The problem is: how should you play? What is your optimal input? After thinking about the problem a bit, you might well start to wonder what sort of access you have to the machine. Can you play it as many times as you want? What are the limits to your access? Well, there a number of scenarios you might consider. (a) Suppose that you are allowed to pull the handle a total of times. What value of will maimize your profit (total output total input)? (b) Suppose that you are allowed to put in a total of $. What value of will maimize your profit (total output total input)? (c) As in (b), you are allowed to put in a total of $. However, every time you pull the handle you have to pay a $ ta. What value of will maimize your profit (total output (total input + total ta))? In each case I want a graphical solution. I m looking for a simple construction on the graph which will identify the optimal value of. (d) As in (c), but suppose we allow the ta levied on each pull to be variable. How do you think the optimal value of should vary as the ta varies? Justify your answer on intuitive grounds and then show that your graphical construction from (c) supports your intuition. For eample, an input of $. will yield an average output of $., whereas an input of $. will yield an average output of $.. Pennies are allowed; for eample it turns out that an input of $. yields an average output of $.. When I ask my students: how would you play this game? most of them assume right away that they get one chance to play. It s when you are allowed multiple plays that things get interesting. The question arises, what are the boundary conditions here? There are fundamentally two kinds of limitations, one on the total number of pulls and the other on the total amount of input money. These are imposed in (a) and (b). To begin I ask my students to use their intuition to guess how the four different optimal values should be ordered. Which should be the largest? the smallest? money making machine //

2 Solution. (a) To maimize your total profit on a fied number of pulls you simply want to maimize your profit on each pull. That is, you want to choose to maimize P() = A(). Graphically, this is the point at which the vertical separation between the A-graph and the line y= is a maimum. The point at which this occurs is illustrated in the diagram at the right. It is the point at which the tangent to the A-graph is parallel to the line y=, and this is seen to occur for very close to =, say at = $.. At this value of we get a payoff of just over, say $., for a profit of P =.. = $. per pull. Be sure you understand why the optimal -value is given by the tangent construction. Since the tangent is parallel to the line y= there is a constant vertical distance between the two lines. This is the same as the distance between the curve and the line y= at =., but since the curve lies under the tangent, it is greater than the distance between the curve and the line y= at all other. (b) Now you are allowed to put a fied total amount into the machine, in this case $. Let s calculate your total profit on that $. If you put in at each pull, you ll get pulls, and so your total profit will be total output total input = A( ) A( ) = A ( ) To maimize this we want to maimize. How do we do that graphically? A ( ) Well observe that for any, is the slope of the line drawn from the origin to the point (, A()) on the graph. To maimize this we want the point on the curve at which this line has the largest possible slope, and this is clearly seen to be the point at which the line from the origin is tangent to the graph. This occurs at close to. for an output of.. The answer is that you should put $. into the machine at each pull. slope= f( )/ f () y = f() money making machine //

3 (c) This is the same as (b) ecept with an added $ ta every time the handle is pulled. If you put in at each pull, you ll again get pulls, and your total profit will be total output (total input + ta) A( ) + = [ A( ) ] A( ) = y = f() To maimize this we want to maimize graphically? A( ). How do we do that Can we adapt the construction we used in (b)? Yes we can. Observe A( ) that for any, is the slope of the line drawn from the point (, ) on the y-ais to the point (, A()) on the graph. b slope=( f( )- b)/ f ()-b b To maimize this we want the point on the curve at which this line has the largest possible slope, and again this is clearly seen to be the point at which the line from (, ) is tangent to the graph. This occurs at close to. for an output of close to.. The answer is that you should put $. into the machine at each pull. (d) If the ta on each pull is increased, our intuition tells us that this ought to cause you to reduce the number of pulls and this will in turn mean a larger input for each pull. Thus the optimal value of should be higher for a higher ta. We can see immediately from the diagram that this is the case. The ta determines the point on the y-ais from which the tangent line is drawn. As the ta is increased, the intercept with the y-ais will rise and the tangent will intersect the curve at a higher point. money making machine //

4 Problems.. Suppose the average payoff for the magical money making machine in the Eample of this section is given by the graph at the right. Again, in each case I want a graphical solution. I m looking for a simple construction on the graph which will identify the optimal value of. (a) Suppose that you are allowed to pull the handle a total of times. What value of will maimize your profit (total output total input)? (b) Suppose that you are allowed to put in a total of $. What value of will maimize your profit (total output total input)? (c) As in (b), you are allowed to put in a total of $. However, every time you pull the handle you have to pay a $ ta. What value of will maimize your profit (total output (total input + total ta))? (d) As in (c), but suppose we allow the ta levied on each pull to be variable. Discuss how the optimal value of changes as the ta varies between and $ per pull. Use the graph to help you eplain your solution. *(e) Referring to part (d), sketch a rough graph of the optimal against the ta t, for t. Again use constructions on the graph at the right to help you eplain your solution.. For this problem we ll use the graph of the magical money making machine in the main Eample but we will think about it in a different way. Suppose that the machine works as follows. As soon as you turn it on, the elves who live inside the bo start making gold. In any period of minutes, they make A() mg. of gold and the graph of A against is plotted at the right. As soon as you pull the handle the machine stops and the amount of gold made to that point comes tumbling out. Note that no input of money is required. Suppose that you have purchased hours of the machine s time. How should you use that time? How much time should you let the machine run before pulling the handle? We consider two different scenarios. In each case use a simple construction on the graph to identify the optimal value of. (a) Suppose that when you pull the handle the machine resets instantly and the elves start work immediately on the new batch of gold. (b) Suppose there is a -minute rest period after the handle is pulled before the elves start work again. Unfortunately, each of these -minute intervals has to count as part of your hours. The first minute is a set-up period in which the elves sharpen their tools. In part (b) the set-up period begins after the -minute rest period. money making machine //

5 . The A- graph of the Eample is made from a piece of the graph k/ for some k. More precisely, if I cut out a piece of the graph k/ and translate it and possibly flip it upside-down, I could lay it eactly along the A- graph. Use this to find an equation for the A- graph.. Look back at problem. What sort of function did I use to draw the graph for this problem (redrawn at the right)? Does it remind you a bit of the sin function? If so you are right: I used a version of sin. In fact I used the equation: y =. sin(. ) +. Describe the effect of each of the four embellishments to the sin-graph on the curve, making reference to the given graph.. A bird spends its time foraging among patches of berries. When it finds a patch, it stays there until the patch is fairly well depleted, and then it goes off in search of a new patch. What it wants to know is how well it should "pick the patch over." If it stays too long in a patch it will be spending valuable foraging time looking for the hidden berries that remain, but, on the other hand, when it leaves the patch, it has to spend some time finding another. When should it leave the patch and go off to find another? What do we need to know to solve the bird's problem? First we need the net amount of food energy E (joules) it gets from a patch if it stays eactly t minutes. This is graphed at the right. Secondly we need to know the amount of time and energy required to find a new patch once the bird decides to leave. Let's suppose that it takes minutes to locate each new patch. Suppose that the amount of energy epended in this search is (a): negligible (b): joules (mainly in flight). In each case, how long should the bird spend in a single patch to maimize its overall intake of energy over a day of foraging? [Note that this is the same as maimizing its energy gain per minute averaged over the day.] Solve the problem with a geometric construction on the above graph. E For small t the graph is nearly linear, suggesting a constant rate of intake of food. But as t increases, the patch becomes depleted, food items are harder to find, and the rate of food intake decreases. Thus the graph is concavedown. t money making machine //