A man wants to build a rectangular pen in which to keep his dog. If he uses one side of his barn, what is the maximum area that he can enclose with 80 meter of fencing?
I just did this. Please don't repeat questions.
To find the maximum area the man can enclose with 80 meters of fencing, we need to consider the shape that would use up the entire fencing. In this case, it would be a rectangle since the man wants to build a rectangular pen.
Let's denote the length of the rectangular pen as "L" and the width as "W". Since one side of the barn is being used, the length of the rectangle will be the same as the width of the barn.
We know that the perimeter of the rectangle is equal to the sum of all sides:
Perimeter = 2L + 2W
Since one side of the barn is used as one length of the rectangle, we can write the equation as:
80 = L + 2W
Next, we need to express the area of the rectangle in terms of L and W. The formula for the area of a rectangle is:
Area = Length × Width
In this case, we can write it as:
Area = L × W
Now, we can express one of the variables in terms of the other. From the first equation, we can solve for L:
L = 80 - 2W
Substituting this into the area formula, we have:
Area = (80 - 2W) × W
To find the maximum area, we need to find the value of W that maximizes the Area.
Taking the derivative of the area function with respect to W:
d(Area)/dW = 80 - 4W
Setting this derivative equal to zero to find the critical point:
80 - 4W = 0
4W = 80
W = 20
Now, we can substitute the value of W back into the equation for L:
L = 80 - 2W
L = 80 - 2(20)
L = 80 - 40
L = 40
So, the maximum area that the man can enclose with 80 meters of fencing is:
Area = L × W = 40 × 20 = 800 square meters.