The Save the World club at a local university is looking to raise money through a bake sale outside of the student center. Out of a desire for efficient fund-raising, they consult with economics majors in the club about the best bake-sale strategies. The econ majors do some reading on behavioral economics and find an interesting result: a wider variety of baked goods increases the chance that any person will stop at their table and peruse the pastries. However, because of the phenomenon of too many choices, the more options that are available, the lower the probability a person will actually buy a baked good (for simplicity we will assume that no one buys more than one baked good). Specifically, the probability that any person walking by will stop and peruse the pastries is given by max(x28,1), where x is the number of different pastries available. If someone stops and looks, the probability that the person will purchase a pastry is given by max(1−2x84,0).

 1st attempt Part 1 (3 points)See Hint How many varieties of pastries should the Save the World club produce to maximize sales? 

Part 2 (3 points)See Hint Suppose that 1000 people stop at the table to look at the pastries. How many pastries will the club sell?

 Part 3 (3 points)See Hint Just before the bake sale, the club president comes to the club with a different opportunity: there are 500 graduating seniors who will be required to attend graduation practice, in which they will sit for at least 2 hours listening to names being called. The club president has procured an opportunity for the Save the World club to have a table there in a location where all 500 seniors are guaranteed to stop. If the club elects to do the bake sale at graduation practice rather than outside the student center, how many different types of pastries should the club make?