A Veterinarian has 80 feet of fence and he wants to enclose a rectangular dog-run along the back side of his office building. He will not fence the side along the building. What are the dimensions of the dog-run that give the maximum area he desires?
40x20
To find the dimensions of the dog-run that give the maximum area, we can use the concept of optimization.
Let's suppose the length of the dog-run is L and the width is W. Since the side along the building does not need to be fenced, there are three sides that need to be fenced (two lengths and one width).
The perimeter of the dog-run can be given by:
Perimeter = 2L + W
According to the problem, the Veterinarian has 80 feet of fence in total, so we can set up an equation:
2L + W = 80
We need to express the area of the dog-run in terms of one variable. The area of a rectangle is given by:
Area = Length * Width
We can write the area as a function of L or W. Let's write it as a function of L:
Area = L * (80 - 2L)
Now, we have the area as a function of the length of the dog-run. To find the maximum area, we can take the derivative of the area function with respect to L and set it equal to zero:
d(Area)/dL = 0
Let's differentiate the area function:
d(Area)/dL = 80 - 4L
Now, set it equal to zero and solve for L:
80 - 4L = 0
4L = 80
L = 20
So, we have found the length of the dog-run, which is 20 feet.
To find the width, substitute the value of L into the perimeter equation:
2L + W = 80
2(20) + W = 80
40 + W = 80
W = 40
Therefore, the dimensions of the dog-run that give the maximum area are a length of 20 feet and a width of 40 feet.