Chapter 3.3, Problem 7CYU

(a)

To determine

To write: the inequalities that represent the production time and deck finishing and quality control time.

Answer to Problem 7CYU

The inequalities are $1.5x+y\le 85$ and $2x+0.5y\le 40$ .

Explanation of Solution

Given Information:

The total number of worker hours per day available for the production = $85$ hours.

The total number of hours available for finishing decks and quality control = $40$ hours.

The table for the skateboard manufacturing time is as follows.

Skateboard manufacturing Time
Board Type	Production Time	Deck finishing/Quality Control Time
Pro Boards	$1.5$ hours	$2$ hours
Speciality Boards	$1$ hour	$0.5$ hours

Let $x$ be the number of pro boards manufactured.

Let $y$ be the number of speciality boards manufactured.

According to constraints production time, write the inequality as follows.

The constraint on total production time= $1.5x+y\le 85$

According to deck finishing and quality control time, write the inequality as follows.

The constrain on total deck finishing time = $2x+0.5y\le 40$

Also the non-negative constraints are $x\ge 0$ and $y\ge 0$

(b)

To determine

To draw: the graph showing the feasible region.

Explanation of Solution

Given Information:

According to constraints , the inequality as follows.

The constraint on total production time is $1.5x+y\le 85$

The constraint on total deck finishing time is $2x+0.5y\le 40$

Also the non-negative constraints are $x\ge 0$ and $y\ge 0$

Explanation for plotting the graph :

The graph of $1.5x+y\le 85$ is plotted as follows

To plot $1.5x+y\le 85$ , first plot the line $1.5x+y=85$ andmark the appropriate area

The graph of $2x+0.5y\le 40$ is plotted as follows

To plot $2x+0.5y\le 40$ , first plot the line $2x+0.5y=40$ mark the appropriate area

Plotting the Graph:

Plot the graph of the inequality $1.5x+y\le 85$ as follows

  

The graph of inequality $1.5x+y\le 85$

On the same graph plot the inequality $2x+0.5y\le 40$

  

The graph of the inequality $2x+0.5y\le 40$

Plot the non-negative constraints $x\ge 0$ and $y\ge 0$

  

(c)

To determine

To list the coordinates of the vertices of the feasible region.

Answer to Problem 7CYU

The coordinates of the vertices of the feasible region are $\left(0,0\right)$ , $\left(0,80\right)$ and $\left(20,0\right)$ .

Explanation of Solution

Given Information:

The feasible region is given as follows

  

The graph of the feasible region

Mark the vertices of the feasible region and write its coordinates.

  

The graph showing the coordinates of the feasible region.

The coordinates of the vertices of the feasible region are $\left(0,0\right)$ , $\left(0,80\right)$ and $\left(20,0\right)$ .

(d)

To determine

To write the function for the total profit on the skateboards.

Answer to Problem 7CYU

Function $f\left(x\right)$ for the profit is given by $f\left(x\right)=50x+65y$

Explanation of Solution

Given Information:

The profit on one pro board = $\$50$

The profit on specialty board = $\$65$

Let $x$ be the number of pro boards manufactured.

Let $y$ be the number of speciality boards manufactured.

The profit on $x$ pro boards = $50x$

The profit on speciality board = $65y$

Therefore, function $f\left(x\right)$ for the profit is given by $f\left(x\right)=50x+65y$

(e)

To determine

To determine the number of each type of skateboards that need to made to have maximum profit. Also find the maximum profit.

Answer to Problem 7CYU

The number of specialty skateboards = $80$

The value of maximum profit = $\$5200$

Explanation of Solution

Given Information:

The profit function is given by $f\left(x\right)=50x+65y$

The coordinates of the vertices are $\left(0,0\right)$ , $\left(0,80\right)$ and $\left(20,0\right)$ .

To maximise the profit function find the value at the vertices of the feasible region as follows.

Vertices	Value of the profit function   $f\left(x\right)=50x+65y$
$\left(0,0\right)$	$50\left(0\right)+65\left(0\right)=0$
$\left(0,80\right)$	$50\left(0\right)+65\left(80\right)=5200$
$\left(20,0\right)$	$50\left(20\right)+65\left(0\right)=1000$

The maximum value is $5200$ . Therefore, to maximise the profit $80$ specialty boards need to be manufactured.

Result:

The number of specialty skateboards = $80$

The value of maximum profit = $\$5200$