Chapter 3, Problem 15PT

a.

To determine

To describe: The system of inequalities that represent the number of chairs and tables.

Answer to Problem 15PT

The system of inequalities that represent the number of chairs and tables is

  $\begin{array}{c}t\ge 0\\ c\ge 0\\ 3c+2t\le 108\\ 0.5c+t\le 20\\ f\left(c,t\right)=15c+35t\end{array}$

Explanation of Solution

Given information:

The processing to make a table and chair is provided below,

  $\begin{array}{ccc}\text{Product}& \begin{array}{l}\text{Carpentry}\\ \text{Time }\left(hr\right)\end{array}& \begin{array}{l}\text{Finishing}\\ \text{Time }\left(hr\right)\end{array}\\ \text{chair}& 3& 0.5\\ \text{table}& 2& 1\end{array}$

108 hours of carpentry and 20 hours of finishing per day are available. The profit for a table is $35 and for chair is $25.

Calculation:

Consider the provided information thatprocessing to make a table and chair is provided below,

  $\begin{array}{ccc}\text{Product}& \begin{array}{l}\text{Carpentry}\\ \text{Time }\left(hr\right)\end{array}& \begin{array}{l}\text{Finishing}\\ \text{Time }\left(hr\right)\end{array}\\ \text{chair}& 3& 0.5\\ \text{table}& 2& 1\end{array}$

108 hours of carpentry and 20 hours of finishing per day are available. The profit for a table is $35 and for chair is $25.

Let c denote the number of chairs and y denote the number of tables.

The constraints are,

  $\begin{array}{c}t\ge 0\\ c\ge 0\\ 3c+2t\le 108\\ 0.5c+t\le 20\end{array}$

The objective function is,

  $f\left(c,t\right)=15c+35t$

b.

To determine

To graph: The region that depicts the possible number of chair and table manufactured.

Explanation of Solution

Given information:

The system of inequalities that represent the number of chairs and tables is

  $\begin{array}{c}t\ge 0\\ c\ge 0\\ 3c+2t\le 108\\ 0.5c+t\le 20\\ f\left(c,t\right)=15c+35t\end{array}$

Graph:

Consider the provided information system of inequalities that represent the number of chairs and tables is

  $\begin{array}{c}t\ge 0\\ c\ge 0\\ 3c+2t\le 108\\ 0.5c+t\le 20\\ f\left(c,t\right)=15c+35t\end{array}$

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 number of chairs and tables to maximize the profit.

Answer to Problem 15PT

The maximum profit is $700 which is when no chair and 20 tables are made.

Explanation of Solution

Given information:

The processing to make a table and chair is provided below,

  $\begin{array}{ccc}\text{Product}& \begin{array}{l}\text{Carpentry}\\ \text{Time }\left(hr\right)\end{array}& \begin{array}{l}\text{Finishing}\\ \text{Time }\left(hr\right)\end{array}\\ \text{chair}& 3& 0.5\\ \text{table}& 2& 1\end{array}$

108 hours of carpentry and 20 hours of finishing per day are available. The profit for a table is $35 and for chair is $25.

Calculation:

Consider the provided information thatprocessing to make a table and chair is provided below,

  $\begin{array}{ccc}\text{Product}& \begin{array}{l}\text{Carpentry}\\ \text{Time }\left(hr\right)\end{array}& \begin{array}{l}\text{Finishing}\\ \text{Time }\left(hr\right)\end{array}\\ \text{chair}& 3& 0.5\\ \text{table}& 2& 1\end{array}$

108 hours of carpentry and 20 hours of finishing per day are available. The profit for a table is $35 and for chair is $25.

Let c denote the number of chairs and y denote the number of tables.

The constraints are,

  $\begin{array}{c}t\ge 0\\ c\ge 0\\ 3c+2t\le 108\\ 0.5c+t\le 20\end{array}$

The objective function is,

  $f\left(c,t\right)=15c+35t$

Plot the above inequalities on coordinate plane.

  

The profit function that is the cost function that is to be maximized is $f\left(c,t\right)=15c+35t$ .

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

Substitute $\left(0,0\right)$ in the function $f\left(c,t\right)=15c+35t$ .

  $f\left(0,0\right)=0$

Substitute $\left(0,20\right)$ in the function $f\left(c,t\right)=15c+35t$ .

  $f\left(0,20\right)=700$

Substitute $\left(36,0\right)$ in the function $f\left(c,t\right)=15c+35t$ .

  $f\left(36,0\right)=540$

Substitute $\left(34,3\right)$ in the function $f\left(c,t\right)=15c+35t$ .

  $f\left(34,3\right)=600$

Since, maximum profit is $700 which is when no chair and 20 tables are made.