Chapter 6.6, Problem 28PPE

To determine

To determine the graph showing possible dimension of the garden.

Explanation of Solution

Given:

Width at least $5$ feet

Length at least $40$ feet

  $150$ feet of fencing only

Formula used:

Perimeter of rectangle $=2\times \left(\text{Long}+\text{Width}\right)$

Let’s consider $x$ and $y$ be the width and length of the garden respectively.

The width of the fencing is at least $5$ feet so $x\ge 5$ . The boundary line of the inequality equation is,

  $\Rightarrow x=5$ which is a vertical line through $5$ on the $x$ axis.

Representing the inequation $x\ge 5$ as a solid line since the inequality has an equal symbol.

  

The boundary line lies on first quadrant so $x$ and $y$ can’t be negative.

The shade is on right as $\ge$ symbol is present.

The length of the fencing is at least $40$ feet so $y\ge 40$ . The boundary line of the inequality equation is,

  $\Rightarrow y=40$ which is a horizontal line passing through $40$ on the $y$ axis.

Representing the equation as a solid line since the inequality has an equal symbol.

  

The boundary line lies on first quadrant so $x$ and $y$ can’t be negative.

The shade is below the boundary line as inequality equation is having $\ge$ symbol.

The total amount of fencing is the perimeter of the dog run which can be represented as,

  $\Rightarrow \text{perimeter}=2\times \left(\text{Long}+\text{Width}\right)$

  $\Rightarrow \text{perimeter}=2\times \left(x+y\right)$

As allowed value of fencing is $150$ feet

So,

  $\Rightarrow 2\times \left(x+y\right)\le 150$$\left[\because \text{perimeter}=2\times \left(x+y\right)\right]$

Dividing both sides by 2,

  $\Rightarrow \left(x+y\right)\le 75$

Subtracting $x$ on both sides,

  $\Rightarrow \left(y\right)\le 75-x$

This line has a slope $-1$ and $y$ intercept of $75$ .

Representing the equation as a solid line since the inequality has an equal symbol.

The boundary line lies on first quadrant so $x$ and $y$ can’t be negative.

The shade is below the boundary line as inequality equation is having $\le$ symbol.

Graph:

  

The variable used is $x$ and $y$ and it represents and length of the garden respectively.

Three inequalities are needed $x\ge 5$ , $y\ge 40$ and $2\times \left(x+y\right)\le 150.$

Interpretation:

Therefore, graph showing possible dimension of the dog run.