Find the Dimensions of a rectangular garden of the greatest area that can be enclosed in 40 feet of fencing.

To find the dimensions of a rectangular garden with the greatest area that can be enclosed in 40 feet of fencing, we need to follow these steps:

Step 1: Understand the problem
We are given that the garden is rectangular and has a perimeter of 40 feet. We want to find the dimensions of the garden that maximize its area.

Step 2: Define the variables
Let's assume the length of the garden is "L" and the width is "W". These will be the variables we'll use to find the dimensions.

Step 3: Formulate the equations
A rectangular garden's perimeter is given by the equation: Perimeter = 2(L + W). We know that Perimeter = 40, so we can write: 2(L + W) = 40.

Step 4: Solve the equation
Now, we can solve the equation to find either L or W in terms of the other variable. Let's solve for W:
2(L + W) = 40
L + W = 20
W = 20 - L

Step 5: Express the area in terms of one variable
The area of a rectangle is given by the equation: Area = L * W. Since we have expressed W in terms of L, we can rewrite the equation as: Area = L * (20 - L).

Step 6: Maximizing the area
To maximize the area, we need to find the critical points where the derivative of the area equation equals zero. Let's differentiate the area equation with respect to L:
d(Area)/dL = 20 - 2L

Setting this derivative equal to zero and solving for L, we get:
20 - 2L = 0
2L = 20
L = 10

So, the length of the garden is 10 feet.

Step 7: Find the width
Using the width equation W = 20 - L, we substitute L = 10:
W = 20 - 10
W = 10

So, the width of the garden is also 10 feet.

Therefore, the dimensions of the rectangular garden with the greatest area that can be enclosed in 40 feet of fencing are 10 feet by 10 feet.

A square 10 by 10 feet has an area of 100 square feet.

A rectangle 15 by 5 feet has an area of 75 sq. ft.