Chapter 2.7, Problem 23E

(a)

To determine

To find: The most Mr. Perez can earn performing these services.

Answer to Problem 23E

The explanation is given below.

Explanation of Solution

Given:

Mr. Perez has an auto repair shop. He offers two bargain maintenance services an oil change and a tune-up. His profit is $12 on an oil change and $20 on a tune-up. It takes Mr. Perez 30 minutes to do an oil change and 1 hour to do a tune-up. He wants to do at least 25 oil changes per week and no more than 10 tune-ups per week. He can spend up to 30 hours each week on these two services.

Let $x$ be the number of hours spend in oil change.

Let $y$ be the number of hours spend in tune-up.

As he wants to do at least 25 oil changes per week and no more than 10 tune-ups per week. He can spend up to 30 hours each week on these two services, so constraints are given below.

  $\begin{array}{c}\frac{1}{2}x+y\le 30\\ x\ge 25\\ y\ge 10\end{array}$

  $\frac{1}{2}x+y\le 30$ comes from putting the time frames(in minutes) in terms of hours.

The function that represent the maximum of the time spending for both.

  $P\left(x,y\right)=12x+20y$ .

The graph of the inequalities is given below.

  

In order to find the most Mr. Perez can earn performing these services, we need to find the maximum value of the function.

Check the value of $P$ at the corner points.

  $\begin{array}{c}P\left(25,10\right)=12\left(25\right)+20\left(10\right)\\ =500\\ P\left(40,10\right)=12\left(40\right)+20\left(10\right)\\ =680\end{array}$

Further simplified as:

  $\begin{array}{c}P\left(60,0\right)=12\left(60\right)+20\left(0\right)\\ =720\\ P\left(25,0\right)=12\left(25\right)+20\left(0\right)\\ =300\end{array}$

As $\left(60,0\right)\&\left(25,0\right)$ does not allow for any tune-ups.

So the function attains its maximum value at $\left(40,10\right)$ .

Therefore the most Mr. Perez can earn performing these services is $\$680$ .

(b)

To determine

To find: The way the constraints be modified to produce a larger profit.

Answer to Problem 23E

The explanation is given below.

Explanation of Solution

Increasing the amount of oil changes, while decreasing the amount of tune-ups, increase profit.

It that is not a large enough increase, he will have to increase the time spent on both activities.

Therefore the explanation is given above.