We are constructing a fence around a rectangular area in our garden and need to determine all the possible dimensions we can have with our limited amount of fencing. Let's use the guiding questions to find all the possible dimensions.
What Variables Will You Use? What Will They Represent?
For the dimensions we will use the variables l and w.
Length of the garden: l
Width of the garden: w
How Many Inequalities Do You Need to Write?
We will use three inequalities to represent the given scenario. One will show the total restrictions of the perimeter, and one for the restrictions on each dimension. Since we know that there is only 150 feet of fencing that can be used, we know the perimeter P must be less than or equal to 150.
P ≤ 150
⇕
2 l+2 w≤150
We are given two other limitations on the garden. The first one is that we want the length to be at least 40 feet. The second is that we want the width to be at least 5 feet. We can create the following inequalities.
l ≥ 40 and w ≥ 5
We now have three inequalities that we can graph as a system.
2 l+2 w≤150 & (I) w ≥ 5 & (II) l ≥ 40 & (III)
We can create the boundary line for Equation (I) by isolating the w-variable in the equation 2 l+2 w=150.
We can now graph the boundary line. Since the inequality is less than or equal to we will use a solid line. When graphing we keep in mind that neither l nor w can take negative values.
Now we can determine where to shade using a test point. We will use the point ( 0, 0) for simplicity.
Since the point makes the inequality true, we will shade below to include it in the solution.
We will follow the same process to graph the next two inequalities. The second inequality, w ≥ 5, will be a horizontal line. We will use the same test point to determine where we will shade.
Since ( 0, 0) made the inequality false, we will shade away from the test point.
Finally, we will now graph the third inequality, l ≥ 40. This inequality will be a vertical line at l= 40. One last time, we will use the same test point to determine where we will shade.
