brett wants to use 40 feet of twine to mark off a rectangular area in his yard for a garden the length ans width for garden with the greatest area?

The greatest area is a square.

brett wants to use 40 feet of twine to mark off a rectangular area in his yard for a garden the length ans width for garden with the greatest area?

To determine the length and width of the garden that would maximize its area using 40 feet of twine, we need to consider the mathematical relationships involved. Let's break it down step by step:

Step 1: Understanding the problem
Brett wants to use 40 feet of twine to mark off a rectangular area in his yard for a garden. The goal is to find the dimensions (length and width) that will result in the greatest possible area.

Step 2: Identifying the variables
Let's assign variables to represent the length and width of the rectangular garden. We'll use "L" for length and "W" for width.

Step 3: Setting up equations
Since we have two variables, we need two equations to represent the given information.

The perimeter equation:
The perimeter of a rectangle is given by the formula P = 2L + 2W.
In this case, the perimeter is given as 40 feet, so we can write:
2L + 2W = 40

The area equation:
The area of a rectangle is given by the formula A = L * W.
We want to find the dimensions that maximize the area, so we'll express this as:
A = L * W

Step 4: Simplifying the equations
Since we want to express "A" in terms of a single variable, we can solve the perimeter equation for one variable and substitute it into the area equation.

From the perimeter equation:
2L + 2W = 40
Divide both sides by 2:
L + W = 20

Solve for L in terms of W:
L = 20 - W

Substituting this into the area equation:
A = (20 - W) * W

Step 5: Maximizing the area
To find the dimensions that maximize the area, we can find the maximum value of the area equation by finding the vertex of the quadratic equation.

The area equation can be rewritten in the form:
A = -W^2 + 20W

Since the coefficient of the W^2 term is negative, the graph of this equation will be an inverted parabola. The vertex of this parabola represents the maximum area.

To find the value of W at the vertex, use the formula:
W_vertex = -b / (2a)

In this case, a = -1 and b = 20.
W_vertex = -20 / (2 * -1)
W_vertex = -20 / -2
W_vertex = 10

Step 6: Finding the length
We already know that L = 20 - W.
Substituting W_vertex into this equation:
L = 20 - 10
L = 10

Step 7: Conclusion
The length (L) and width (W) of the rectangular garden that will result in the greatest area with 40 feet of twine is 10 feet and 10 feet, respectively.

Therefore, Brett should make his garden a square with each side measuring 10 feet.