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.