Chapter 3.4, Problem 7CYU

a.

To determine

To describe: The system of inequalities that represent the number of pro boards and specialty boards.

Answer to Problem 7CYU

The system of inequalities that represent the number of pro boards and specialty boardsis

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

Explanation of Solution

Given information:

A table that depicts time required to manufacture skateboards.

  $\begin{array}{ccc}\text{Board Type}& \text{Production Time}& \text{Deck Finishing and Quality Control}\\ \text{Pro Boards}& 1.5\text{ hours}& 2\text{ hours}\\ \text{Specialty Boards}& 1\text{ hour}& 0.5\text{ hours}\end{array}$

85 hours are available for production time and 40 hours are available for deck finishing and quality control.

Calculation:

Consider the provided information thata table that depicts time required to manufacture skateboards.

  $\begin{array}{ccc}\text{Board Type}& \text{Production Time}& \text{Deck Finishing and Quality Control}\\ \text{Pro Boards}& 1.5\text{ hours}& 2\text{ hours}\\ \text{Specialty Boards}& 1\text{ hour}& 0.5\text{ hours}\end{array}$

85 hours are available for production time and 40 hours are available for deck finishing and quality control.

Let x denote the number of pro boards manufactured and y denote the number of specialty boards manufactured.

1.5 hours for pro and 1 hour for specialty boards as production time each is required. Maximum 85 hours are available for production time is available.

Similarly, 2 hours for pro and 0.5 hours for specialty boards as deck finishing and quality control each is required. Maximum 40 hours are available for deck finishing and quality control.

The constraints are,

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

b.

To determine

To graph: The region that depicts the represent the number of pro boards and specialty boards.

Explanation of Solution

Given information:

The system of inequalities that represent the number of pro boards and specialty boards is

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

Graph:

Consider the provided information system of inequalities that represent the number of pro boards and specialty boards is

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

Plot the above inequalities on coordinate plane.

  

Interpretation:

The shaded region represents the feasible. It is a quadrilateral with 4 corner points.

c.

To determine

To calculate:The coordinates of feasible region that represent the number of pro boards and specialty boards.

Answer to Problem 7CYU

The coordinates of feasible region are $\left(0,0\right),\left(20,0\right)\text{ and }\left(0,80\right)$ .

Explanation of Solution

Given information:

The system of inequalities that represent the number of pro boards and specialty boards is

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

Consider the provided information system of inequalities that represent the number of pro boards and specialty boards is

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

Plot the above inequalities on coordinate plane.

  

The shaded region represents the feasible. It has3 corner points. The coordinates of feasible region are $\left(0,0\right),\left(20,0\right)\text{ and }\left(0,80\right)$ .

d.

To determine

To calculate:The profit function when skateboards are manufactured.

Answer to Problem 7CYU

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

Explanation of Solution

Given information:

A table that depicts time required to manufacture skateboards.

  $\begin{array}{ccc}\text{Board Type}& \text{Production Time}& \text{Deck Finishing and Quality Control}\\ \text{Pro Boards}& 1.5\text{ hours}& 2\text{ hours}\\ \text{Specialty Boards}& 1\text{ hour}& 0.5\text{ hours}\end{array}$

85 hours are available for production time and 40 hours are available for deck finishing and quality control.

Calculation:

Consider the provided information thata table that depicts time required to manufacture skateboards.

  $\begin{array}{ccc}\text{Board Type}& \text{Production Time}& \text{Deck Finishing and Quality Control}\\ \text{Pro Boards}& 1.5\text{ hours}& 2\text{ hours}\\ \text{Specialty Boards}& 1\text{ hour}& 0.5\text{ hours}\end{array}$

85 hours are available for production time and 40 hours are available for deck finishing and quality control.

Let x denote the number of pro boards manufactured and y denote the number of specialty boards manufactured.

1.5 hours for pro and 1 hour for specialty boards as production time each is required. Maximum 85 hours are available for production time is available.

Similarly, 2 hours for pro and 0.5 hours for specialty boards as deck finishing and quality control each is required. Maximum 40 hours are available for deck finishing and quality control.

The constraints are,

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

Since, company makes a profit of $50 for each pro board and $65 for specialty board.

The objective function or profit function is,

  $f\left(x,y\right)=50x+65y$

e.

To determine

To calculate:The number of pro and specialty boards to be manufactured to maximize the profit.

Answer to Problem 7CYU

The maximum profit is $\$5200$ when 80 specialty boards are manufactured.

Explanation of Solution

Given information:

A table that depicts time required to manufacture skateboards.

  $\begin{array}{ccc}\text{Board Type}& \text{Production Time}& \text{Deck Finishing and Quality Control}\\ \text{Pro Boards}& 1.5\text{ hours}& 2\text{ hours}\\ \text{Specialty Boards}& 1\text{ hour}& 0.5\text{ hours}\end{array}$

85 hours are available for production time and 40 hours are available for deck finishing and quality control.

Calculation:

Consider the provided information that a table that depicts time required to manufacture skateboards.

  $\begin{array}{ccc}\text{Board Type}& \text{Production Time}& \text{Deck Finishing and Quality Control}\\ \text{Pro Boards}& 1.5\text{ hours}& 2\text{ hours}\\ \text{Specialty Boards}& 1\text{ hour}& 0.5\text{ hours}\end{array}$

85 hours are available for production time and 40 hours are available for deck finishing and quality control.

Let x denote the number of pro boards manufactured and y denote the number of specialty boards manufactured.

1.5 hours for pro and 1 hour for specialty boards as production time each is required. Maximum 85 hours are available for production time is available.

Similarly, 2 hours for pro and 0.5 hours for specialty boards as deck finishing and quality control each is required. Maximum 40 hours are available for deck finishing and quality control.

The constraints are,

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ 1.5x+y\le 85\\ 2x+0.5y\le 40\end{array}$

Since, company makes a profit of $50 for each pro board and $65 for specialty board.

The objective function or profit function is,

  $f\left(x,y\right)=50x+65y$

Plot the above inequalities on coordinate plane.

  

The shaded region represents the feasible. The coordinates of feasible region are $\left(0,0\right),\left(20,0\right)\text{ and }\left(0,80\right)$ .

Substitute the vertices of the feasible region to find the point at which maximum profit is there.

Substitute $\left(0,0\right)$ in the function $f\left(x,y\right)=50x+65y$ .

  $f\left(0,0\right)=0$

Substitute $\left(20,0\right)$ in the function $f\left(x,y\right)=50x+65y$ .

  $f\left(20,0\right)=1000$

Substitute $\left(0,80\right)$ in the function $f\left(x,y\right)=50x+65y$ .

  $f\left(0,80\right)=5200$

Thus, the maximum profit is $\$5200$ when 80 specialty boards are manufactured.