Jim is setting up a rectangular dog run in
his backyard. He buys six 3-foot sections
of fencing and a 3-foot wide gate. What
are the dimensions of the dog run with
the greatest area in each situation?
a) Jim uses the yard fence for one side of
the dog run.
b) Jim uses the corner of the yard fence
for two sides of the dog run.
To find the dimensions of the dog run with the greatest area, we need to consider two situations: when Jim uses the yard fence for one side of the dog run, and when Jim uses the corner of the yard fence for two sides of the dog run.
a) When Jim uses the yard fence for one side of the dog run:
Let's assume the width of the dog run (parallel to the yard fence) is x feet. The length of the dog run (perpendicular to the yard fence) is 3 feet more than x, as it includes the 3-foot wide gate.
Therefore, the dimensions of the dog run are:
Width = x feet
Length = x + 3 feet
To find the area of the dog run, we multiply the width (x) by the length (x + 3):
Area = Width × Length = x(x + 3)
b) When Jim uses the corner of the yard fence for two sides of the dog run:
In this situation, the fence forms two adjacent sides of the rectangle. Let's assume one side of the dog run (perpendicular to the yard fence) is x feet. The other side (parallel to the yard fence) will be 3 feet longer to accommodate the 3-foot wide gate.
Therefore, the dimensions of the dog run are:
Width = 3 feet
Length = x + 3 feet
To find the area of the dog run, we multiply the width (3) by the length (x + 3):
Area = Width × Length = 3(x + 3)
Now we can find the dimensions of the dog run with the greatest area by maximizing the area formulas for each situation.
For situation (a), we can expand the area equation and find the maximum by differentiating it with respect to x, setting the derivative equal to zero, and solving for x.
For situation (b), we can also expand the area equation and find the maximum in a similar manner.
The largest area is always a square.
18 feet plus the 3 foot gate = 21 feet. He needs that evenly divided for 3 sides so I would 7 feet on each side. However, the problem is that they come in 3 foot sections. So you have to work this out so that you get as close to a square as possible.