Mr. Jones needs to build a rectangular fence in his backyard to keep their dog safe. The length of the fence (y) should be at least 60 ft, and the distance around should be no more than 260 ft. Which system of inequalities and graph represent the possible dimensions of the pen?
Let's assume that the width of the fence is x ft.
The given information can be translated into the following inequalities:
1. Length of the fence should be at least 60 ft: y ≥ 60
2. Distance around the fence should be no more than 260 ft: 2x + 2y ≤ 260
So, the system of inequalities that represents the possible dimensions of the pen is:
y ≥ 60
2x + 2y ≤ 260
To graph the system, we can first graph the boundary lines for each inequality, and then shade the region that satisfies both inequalities.
Graphing y ≥ 60:
Since y should be at least 60, we can plot a horizontal line at y = 60.
Graphing 2x + 2y ≤ 260:
We can convert the inequality into slope-intercept form:
2x + 2y ≤ 260
→ y ≤ (260 - 2x)/2
→ y ≤ 130 - x
To graph the boundary line, we plot the points (0, 130) and (130, 0) and draw a line passing through them.
Now, we can shade the region below the line y = 60 and below the line y = 130 - x.
The shaded region bounded by these two lines represents the possible dimensions of the pen.
Here is a graph that represents the system of inequalities:
┌────────────────────┐
│ │
│ /\ │
│ / \ │
│ / \ │
│ / \ │
│ / \ │
│ ____________\_____│_________
│ │
│ y │ 60
│ │
└────────────────────┘
On the graph, the shaded region should be below the line y = 60 and below the line y = 130 - x.