# Constrained optimization problem

Step by step explanation/solution

Brian is taking three courses this semester: economics, statistics, and finance. He has decided to spend 19 hours per week studying (in addition to attending all his classes) and his objective is to maximize his average grade, which means maximizing the total of his grades in the three courses. The table shows Brian's estimate of the relation between time spend studying and his grade for each course. Notice it is assumed that Brian will spend at least 4 hours per week studying each of the three courses.

Grade in Grade in Grade in

Hours of Study Economics Statistics Finance

4 68 63 64

5 76 72 71

6 83 80 77

7 87 85 83

8 90 88 87

9 92 90 90

10 94 91 92

11 95 92 94

12 96 93 95

13 96 94 95

a. Is this a constrained optimization problem or an unconstrained optimization problem?

b. If Brian is going to spend a total of 19 hours a week studying what is the optimal mix of studying between the three courses? That is, how many hours will he devote to economics? To statistics? To finance?

c. What is the maximum average grade Brian can earn if he studies 19 hours per week?

d. What if Brian decides to spend 25 hours, rather than 19, studying? How does the optimal mix change? How does the maximum average grade change?

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#### Solution Preview

This is a constrained optimization problem because he has a limited amount of time to spend studying.

He will devote hours based on which has the highest marginal gain in score until he has spent 19 hours. Imagine I have only twelve hours to spend, the problem says I spend it ...