Chapter 3.4, Problem 13PPSE

Solution

a.

To find: The constraints for the situations, where $\$2100$ for planting maples and spruce trees and area available for planting $45000f{t}^2.$

The constraints are

  $\begin{array}{l}30x+40y\le 2100\\ 650x+300y\le 45000\\ x\ge 0\\ y\ge 0\end{array}$

Given information:

The given statement is $\$2100$ for planting maples and spruce trees and area available for planting $45000f{t}^2.$ and the data is

  

Concept used:

Linear programming problems represent constraints by equations or inequalities and by system of equations.

The graph of the system is the feasible region and it contains all the points that satisfy all the constraints.

Calculations:

Here maximum amount should $\$2100$ ,so inequality for planting cost will be

  $30x+40y\le 2100$

The maximum area should be $45000f{t}^2.$ so inequality for area is $650x+300y\le 45000$ .

  $\begin{array}{l}x\ge 0\\ y\ge 0\end{array}$

Thus the constraints are

  $\begin{array}{l}30x+40y\le 2100\\ 650x+300y\le 45000\end{array}$

b.

To find: The objective function for the situations, where $\$2100$ for planting maples and spruce trees and area available for planting $45000f{t}^2.$

objective function here is $A=650x+300y$ .

Given information:

The given statement is $\$2100$ for planting maples and spruce trees and area available for planting $45000f{t}^2.$ and the data is

  

Concept used:

Linear programming problems represent constraints by equations or inequalities and by system of equations..

Calculations:

Here carbon dioxide absorption from the spruce tree is $650\text{lb/yr}$ and maple tree is $300\text{lb/yr}$ .

So based on the question objective function is $A=650x+300y$ .

Thus objective function here is $A=650x+300y$ .

c.

To find: the vertices and graph the region the constraints are

  $\begin{array}{l}30x+40y\le 2100\\ 650x+300y\le 45000\\ x\ge 0\\ y\ge 0\end{array}$

The vertices are $\left(0,52.5\right),\left(69,0\right)$ .

Given information:

The constraints are

  $\begin{array}{l}30x+40y\le 2100\\ 650x+300y\le 45000\\ x\ge 0\\ y\ge 0\end{array}$

Concept used:

The graph of the system is the feasible region and it contains all the points that satisfy all the constraints.

Plot the graph:

The graph is given below:

  

And the vertices are $\left(0,52.5\right),\left(69,0\right)$ .

d.

To find: the maximum of the function $A=650x+300y$ at the coordinate points of the vertex are $\left(0,52.5\right),\left(69,0\right)$ .

  $A$ has a maximum value of $44850$ at $\left(69,0\right).$

The values of $x$ and $y$ that has maximum value are $69$ and $0$ respectively.

Given information:

The function is $A=650x+300y$ and the coordinate points of the vertex are $\left(0,52.5\right),\left(69,0\right)$ .

Evaluate $A$ at each vertex.

  $\begin{array}{c}A=650x+300y\\ A\left(0,52.5\right)=650\times 0+300\times 52.5=15750\\ A\left(69,0\right)=650\times 69+300\times 0=44850\end{array}$

  $A$ has a maximum value of $44850$ at $\left(69,0\right).$

The values of $x$ and $y$ that has maximum value are $69$ and $0$ respectively.