Answer :
Maximize area with a square use 500 meters for both sides.
Here's how the landowner can maximize the area of their rectangular region:
1. Perimeter constraint: 2000 meters of fencing will form the remaining three sides of the rectangle. Let "x" be the length of one side and "y" be the length of the other.
The perimeter equation is: 2x + y = 2000.
2. Area maximization: We want to maximize the area, which is calculated as: A = xy.
3. Express one variable in terms of the other: To simplify, we can express y in terms of x from the perimeter equation: y = 2000 - 2x.
4. Substitute and maximize: Replace y in the area equation with the expression: A = x(2000 - 2x).
Now, you have an equation with only one variable (x).
5. Find the critical point: To find the dimensions that maximize the area, take the derivative of A with respect to x, set it equal to zero, and solve for x. This will give you a critical point, which usually indicates the maximum value.
6. Calculate dimensions: Using the critical point value for x, calculate the corresponding value for y using the perimeter equation or the substituted area equation.
Following these steps and solving the equations, you'll find that the maximum area is achieved when both sides are 500 meters long. This utilizes all the 2000 meters of fencing and creates a square, which is known to maximize the area for a given perimeter.
A landowner wishes to use 2000 meters of fencing to enclose a rectangular region. Suppose one side of the property lies along a stream and thus does not need to be fenced in. What should the lengths of the sides be in order to maximize the area?
