Maximizing Garden Area with Fencing Material

What are the dimensions of the garden that maximize its area?

A rectangular garden is to be fenced off using 100 meters of fencing material. What are the dimensions of the garden that maximize its area?

Final Answer:

The dimensions of the rectangular garden that maximize its area are 25 meters in length and 25 meters in width.

To determine the dimensions of the rectangular garden that will result in the maximum area given 100 meters of fencing material, we can use the principles of calculus and optimization. Let's denote the length of the rectangle as 'L' and the width as 'W'.

The perimeter of a rectangle is given by the formula: 2L + 2W = 100. Solving for one of the variables, we get: L = 50 - W

The area of the rectangle is given by the formula: A = L x W. Substituting the value of L from the perimeter equation, we get: A = (50 - W) x W = 50W - W^2.

To find the maximum area, we can take the derivative of the area equation with respect to W, and set it equal to zero: dA/dW = 50 - 2W = 0. Solving for W, we get W = 25. Substituting that back into the equation, we find L = 25.

So, the dimensions of the garden that maximize its area are 25 meters in length and 25 meters in width. This will create a square-shaped garden, which is the optimal configuration for the given amount of fencing material.