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  
  1. 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?

Answer:

When Brian devotes 6 hours each to economics and statistics and 7 hours to finance, the total grades = 83+80+83 = 246

This is the maximum total grade, Brian can achieve out of all possible combinations using 19 hours.

So Brian will devote 6 hours each to economics and statistics and 7 hours to finance in order to maximize its total grade.

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

Answer:

When Brian devotes 6 hours each to economics and statistics and 7 hours to finance, the maximum total grades = 83+80+83 = 246

So maximum average grade = total maximum grade/number of subjects = 246/3 = 82

  • 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?

Answer:

In this case, Brian will devote 8 hours each to economics and statistics and 9 hours to finance. So total maximum grade = 90+88+90 = 268

The new maximum average grade = total maximum grade/number of subject = 268/3 = 89.33

We note that with increase in studying hours, maximum average grade rises.

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