Chapter 4.6, Problem 53E

a.

To determine

To find: The type of answers to earn the highest score.

Answer to Problem 53E

The type of answers to earn highest score is $120$ .

Explanation of Solution

Given:

The time to attempt the questions is one hour and the answers of all the questions are correct.

Calculation:

Let the variable $x$ for short answer and $y$ for essay questions.

Total number of questions are $20$ .

  $x+y\le 20$

And,

It takes two Let the variable $x$ for short answer and $y$ for essay questions.

minutes to answer short answers and takes twelve minutes to answer essay questions. Hence,

  $2x+12y\le 60$

Each short answer question is worth five points. And each essay question is worth fifteen points. Thus,

  $M\left(x,y\right)=5x+15y$ .

Now, the maximum marks are obtained by putting the value in the equation $M\left(x,y\right)=5x+15y$ .

  

  $\begin{array}{c}M\left(0,5\right)=5\left(0\right)+15\left(5\right)\\ =75\end{array}$

For $M\left(18,2\right)$

  $\begin{array}{c}M\left(18,2\right)=5\left(18\right)+15\left(2\right)\\ =120\text{maximum}\end{array}$

For $M\left(20,0\right)$

  $\begin{array}{c}M\left(20,0\right)=5\left(20\right)+15\left(0\right)\\ =100\end{array}$

I did not test $\left(0,0\right)$ as we are looking for maximum score.

Therefore, the type of answers to earn highest score is $120$ .

b.

To determine

To find: The type of answers to earn highest score.

Answer to Problem 53E

The maximum score is $180$ .

Explanation of Solution

Given:

The time to attempt the questions is two hour and the answers of all the questions are correct.

Calculation:

Let the variable $x$ for short answer and $y$ for essay questions.

Total number of questions are $20$ .

  $x+y\le 20$

And,

It takes two Let the variable $x$ for short answer and $y$ for essay questions.

minutes to answer short answers and takes twelve minutes to answer essay questions. Hence,

  $2x+12y\le 60$

Each short answer question is worth five points. And each essay question is worth fifteen points. Thus,

  $M\left(x,y\right)=5x+15y$ .

Now, the maximum marks are obtained by putting the value in the equation $M\left(x,y\right)=5x+15y$ .

  

For $M\left(0,10\right)$ ,

  $\begin{array}{c}M\left(0,10\right)=5\left(0\right)+15\left(10\right)\\ =150\end{array}$

For $M\left(12,8\right)$ ,

  $\begin{array}{c}M\left(12,8\right)=5\left(12\right)+15\left(8\right)\\ =180\text{maximum}\end{array}$

For $M\left(20,0\right)$ ,

  $\begin{array}{c}M\left(20,0\right)=5\left(20\right)+15\left(0\right)\\ =100\end{array}$

Therefore, the maximum score is $180$ .