Final answer:
To maximize the area of a rectangular pen with one side along a river using 100 ft of fencing, the dimensions should be 25 ft for width and 50 ft for length. This is derived from setting up an equation for the perimeter (2w + l = 100), substituting it into the area formula, and finding the width that maximizes the area.
Step-by-step explanation:
To maximize the area of a rectangular pen using only 100 ft of fencing while one side is along a river, we need to determine the dimensions that will give us the largest area. Since one side of the rectangle is along the river, we don't need to use fencing for that side. We only need fencing for the other three sides.
Let's say the width (the side perpendicular to the river) is w feet and the length (the side parallel to the river) is l feet. We have 100 feet of fencing available, so we need to account for only three sides of the rectangle, which gives us the equation:
2w + l = 100
Now, to find the area A, we use the formula:
A = w * l
Substituting the value of l from the earlier equation, we get:
l = 100 - 2w
So the area equation becomes:
A = w * (100 - 2w)
Now, to find the maximum area, we can take the derivative of A with respect to w and set it to zero:
A'(w) = 100 - 4w = 0
Solving for w, we find that:
w = 25 ft
Substitute w back into the equation for l, we get:
l = 100 - 2(25)
l = 50 ft
Therefore, to maximize the area with only 100 ft of fencing, the dimensions should be 25 ft for width and 50 ft for length.