Y5

Consider a module with the following marking scheme: There are two tests and your overall grade is equal to the minimum of your grades on the two tests. So, for example, if you score 88 on test 1 and 34 on test 2, your overall grade in the course is 34.

(a) Suppose that a student who spends no time studying for a test receives zero marks on that test. Suppose that each mark received on test 1 costs p1 minutes of study time. Each mark received on test 2 costs p2 minutes of study time. Consider a student who spends a total of M minutes studying for the two tests and allocates this time most efficiently to achieve the highest overall grade. What will be this student's overall grade? (your answers should depend on M, p1 and p2)

(b) If a student wants to achieve an overall grade of G with the smallest possible total time studying, how much time must the student spend studying on test 1? How much time must the student spend studying on test 2?

(c) Suppose that a student has the utility function, U(G, M) = G A2 M2,

where G is the student's overall grade, M is the number of minutes the student spends studying, and A > 0 is a parameter that reflects how much the student dislikes studying. For this student, find the total amount of study time (M) that maximizes utility. (Hint: use your answer from part (a) and take a first-order condition).

(d) Continuing with part (c): If a student chooses the utility-maximizing amount of study time and allocates this most efficiently between the two tests, then what will be the overall grade in the course? Your answer should depend on the parameters, A, p1 and p2.

(e) Roy has a utility function like the one above, and for him, p1 = 5, p2 = 15 and A = 1/36000 . If Roy chooses his total study time to maximize utility, how many total minutes does he spend studying and what will be his

overall grade in the course?