Chapter 6.6, Problem 3P

To determine

To determine the graph showing possible dimension of the dog run.

Explanation of Solution

Given:

Width at least $10$ feet

Long at most $50$ feet

  $126$ feet of fencing is given

Formula used:

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

Let’s consider $x$ and $y$ be the width and long of the dog run respectively.

The width of the fencing is at least $10$ feet so $x\ge 10$ . The boundary line of the inequality equation is,

  $\Rightarrow x=10$ which is a vertical line through $10$ on the $x$ 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 on right as $\ge$ symbol is present.

The long of the fencing is at most $50$ feet so $y\le 50$ . The boundary line of the inequality equation is,

  $\Rightarrow y=50$ which is a horizontal line passing through $50$ 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 $\le$ 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 $126$ feet

So,

  $\Rightarrow 2\times \left(x+y\right)\le 126$$\left[\because \text{perimeter}=2\times \left(x+y\right)\right]$

Dividing both sides by 2,

  $\Rightarrow \left(x+y\right)\le 63$

Subtracting $x$ on both sides,

  $\Rightarrow \left(y\right)\le 63-x$

This line has a slope $-1$ and $y$ intercept of $63$ .

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:

  

Interpretation:

Therefore, graph showing possible dimension of the dog run.