Chapter 2.7, Problem 15E

To determine

To find: the number of questions from each section the students needs to attempt to get higher score; if not possible state whether the problem is infeasible, unbounded or has alternate optimal solutions.

Answer to Problem 15E

Section I: 10 questions

Section II: 2 questions

Explanation of Solution

Given information:

Let $x$ be the questions of section I and $y$ be the questions of section II.

Maximum Marks of section I = $10x$

Maximum Marks of section II = $15y$

Time constraint for attempting section I = $6x$

Time constraint for attempting section II = $15x$

Total time = 90 mins

Minimum question to be attempted from the section II = 2

Calculation:

Let $M\left(x,y\right)$ be the maximum score.

The inequalities derived from the given information are as follows:

  $\begin{array}{c}M\left(x,y\right)=10x+15y\\ 6x+15y\le 90\\ x\ge 0\left[\text{Assume}\right]\end{array}$

Use a graphing calculator to plot the inequalities as shown below.

  

The vertices are $\left(0,6\right),\left(10,2\right)\text{ and }\left(0,2\right)$ . Now test each vertex as shown below.

  $\begin{array}{c}M\left(0,6\right)=10\left(0\right)+15\left(6\right)\\ =90\\ M\left(10,2\right)=10\left(10\right)+15\left(2\right)\\ =100+30\\ =130\\ M\left(0,2\right)=10\left(0\right)+15\left(2\right)\\ =30\end{array}$

Thus, the number of questions the student needs to attempt from section I and section II to get higher score are 10 and 2 respectively.