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.
a) If Jim uses the yard fence for one side of the dog run, he will have 6 sections of fencing left to create the remaining three sides of the dog run. Since each section is 3 feet long, Jim will have a total of 6 x 3 = 18 feet of fencing left.
To find the dimensions of the dog run with the greatest area, we need to consider that the dog run is rectangular.
Let's say the length of the dog run is L and the width is W.
The length of the dog run will be L = 18 feet, as Jim has 18 feet of fencing left.
To maximize the area, we need to find the width that will yield the greatest possible area. We can express the width in terms of L and solve for the area:
Width = W
Area = L x W = 18 x W
To find the maximum area, we need to differentiate the area function with respect to W and set it to zero:
d(Area) / d(W) = 18 - 0 = 0
Therefore, the width that will yield the greatest area is W = 0. This indicates that the dog run will form a line along the yard fence, with no width. In other words, the dog run will be a straight line.
b) If Jim uses the corner of the yard fence for two sides of the dog run, he will have 6 sections of fencing left to create the other two sides of the dog run.
For this scenario, since the two sides are adjacent to each other and share a corner of the yard fence, they must have the same length. Let's say the length of each side is L.
We have a total of 6 sections of fencing remaining, which is equal to 6 x 3 = 18 feet.
So, the length of each side will be L = 18 / 2 = 9 feet.
Therefore, in this situation, the dog run will have dimensions of 9 feet by 9 feet, resulting in a square dog run.
To find the dimensions of the dog run with the greatest area, we need to consider two scenarios:
a) Jim uses the yard fence for one side of the dog run:
In this situation, let's assume the length of the yard fence is L. Jim uses one section of the fencing for the width of the dog run, and the gate adds an additional 3 feet to the length. Therefore, the length of the dog run is L + 3 feet, and the width is 3 feet.
The formula for the area of a rectangle is: Area = Length * Width.
So, the area of the dog run in this scenario would be: Area = (L + 3) * 3.
b) Jim uses the corner of the yard fence for two sides of the dog run:
In this situation, let's assume the length of the yard fence is L. Jim uses two sections of the fencing to create two adjacent sides of the dog run. The gate will be used to close off one end of the rectangle. Therefore, the two sides of the dog run formed by the fencing has a total length of L + L = 2L, and the width is still 3 feet.
The area of the dog run in this scenario would be: Area = 2L * 3.
Now, to determine which scenario yields the greatest area, we can compare the two formulas.
For scenario a):
Area = (L + 3) * 3
For scenario b):
Area = 2L * 3
To find the maximum area, we can evaluate the two expressions. For scenario a), we need to expand the expression:
Area = 3L + 9
For scenario b), we have:
Area = 6L
Now, we can see that in both scenarios, the area is directly proportional to the length. However, in scenario b), the coefficient multiplying the length (6) is greater than the coefficient in scenario a) (3). Therefore, scenario b) will result in a larger area.
In summary:
- In scenario a), the dog run with the greatest area will have dimensions (L + 3) * 3, with the length being the length of the yard fence and the width being 3 feet.
- In scenario b), the dog run with the greatest area will have dimensions 2L * 3, with the length being the length of the yard fence and the width being 3 feet.