A rectangular field is to be enclosed by 300 feet of fencing. Find the dimensions of the enclosure that maximizes the area.

To find the dimensions of the enclosure that maximize the area, we can use the concept of derivatives and optimization. Let's break down the problem into steps:

Step 1: Define the variables
Let's assume the length of the field is 'x' feet, and the width is 'y' feet.

Step 2: Formulate the equation
The perimeter of the rectangular field is given as 300 feet, so we can write the equation:
2x + 2y = 300

Step 3: Isolate one variable in terms of the other
Solving the equation from Step 2 for 'y' gives:
y = (300 - 2x)/2
y = 150 - x

Step 4: Define the area equation
The area of a rectangle is calculated by multiplying the length and width. In this case, the area 'A' can be expressed as:
A = xy

Step 5: Substitute the value of 'y' from Step 3 into the area equation
Substituting the value of 'y' into the area equation gives:
A = x(150 - x)
A = 150x - x^2

Step 6: Differentiate the area equation
To find the value of 'x' that maximizes the area, we need to find the derivative of 'A' with respect to 'x':
dA/dx = 150 - 2x

Step 7: Set the derivative equal to zero and solve for 'x'
To determine the critical points, we set the derivative equal to zero:
150 - 2x = 0
2x = 150
x = 75

Step 8: Substitute the value of 'x' into the equation for 'y' from Step 3
Substituting the value of 'x' into the equation for 'y', we get:
y = 150 - x
y = 150 - 75
y = 75

Step 9: Calculate the maximum area
To find the maximum area, substitute the values of 'x' and 'y' into the area equation:
A = xy
A = (75)(75)
A = 5625 square feet

Step 10: Determine the dimensions of the enclosure
Based on our calculations, the dimensions that maximize the area are a length of 75 feet and a width of 75 feet.

The largest area would be a square.

300/4 = ?