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.
You must go to a strange school if geometry is called "Sheridan" there. If you want a geometry tutor, say so in your "School Subject" line.
To find the dimensions of the dog run with the greatest area, we need to consider the given constraints and determine the best arrangement of the fencing sections.
a) Jim uses the yard fence for one side of the dog run:
In this case, we have one side of the dog run already secured by the yard fence. Let's assume the length of this side is "L." Since Jim has six 3-foot sections of fencing, he can use the remaining five sections to make up the other three sides of the dog run.
To maximize the area, we should create a square-shaped dog run since a square has the maximum area for a given perimeter. Therefore, the other three sides should also have a length of "L" each.
The perimeter of the dog run will be P = L + 3 + 3 + 3 + 3 = L + 12 feet. Jim has a total of six 3-foot sections, which means he has 6 x 3 = 18 feet of fencing. Since he needs L + 12 feet of fencing for the perimeter, we can set up the equation:
L + 12 = 18
L = 18 - 12
L = 6
Thus, the dimensions of the dog run with the greatest area will be 6 feet by 6 feet.
b) Jim uses the corner of the yard fence for two sides of the dog run:
In this case, Jim can utilize the corner of the yard fence to create two sides of the dog run. Let's assume the length of one side is "L" and the length of the other side is "W." Since Jim has six 3-foot sections of fencing, he can use the remaining four sections to make up the other two sides of the dog run.
To maximize the area, we should aim for a rectangle since it allows for a larger area compared to other shapes with the same perimeter. Therefore, we need to determine the values of "L" and "W" that would maximize the area.
The perimeter of the dog run will be P = L + W + 3 + 3 + 3 + 3 = L + W + 12 feet. Jim has a total of six 3-foot sections, so he has 6 x 3 = 18 feet of fencing. Since he needs L + W + 12 feet of fencing for the perimeter, we can set up the equation:
L + W + 12 = 18
L + W = 18 - 12
L + W = 6
To maximize the area, we need to determine the values of "L" and "W" that satisfy the above equation while also maximizing their product, which represents the area.
One possible combination would be L = 3 and W = 3, which gives us the maximum area of 3 x 3 = 9 square feet.
Therefore, in situation b), the dimensions of the dog run with the greatest area will be 3 feet by 3 feet.