Final answer:
Farmer Joe can achieve the maximum area of 800 square meters by fencing in a rectangle with dimensions 40 meters in length and 20 meters in width, using his available 80 meters of fencing.
Step-by-step explanation:
The question concerns mathematics, specifically related to geometry and optimization. Farmer Joe wants to fence in a rectangular area with one side bordered by a stream, using 80 meters of fencing. Ideally, we're looking to find the dimensions that will give us the maximum enclosed area.
If we let the length of the rectangle parallel to the stream be 'L' and the width be 'W', the perimeter of three sides of the fenced area will be 2W + L = 80 meters, since one length does not require fencing due to the stream.
To find the maximum area, we can set up and maximize the area function A = L * W. Using the perimeter equation to express L in terms of W, we get L = 80 - 2W. Substituting this into the area function gives us A = W * (80 - 2W). Differentiating and finding the critical points, we conclude that the maximum area occurs when W = 20 meters, which implies L = 40 meters (since 80 - 2*20 = 40).
Thus, the correct answer is option A: Length: 40 meters, Width: 20 meters, Maximum Area: 800 square meters.