A farmer will make a rectangular pen with 100 feet of fence using pasrt of his barn for one side of the pen. What is the largest area he can enclose?

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Just change the numbers.

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To find the largest area the farmer can enclose with 100 feet of fence, we need to determine the dimensions of the rectangular pen that would maximize the area. Let's break down the problem step by step:

Step 1: Set up the problem
Let's assume the dimensions of the rectangular pen are length (L) and width (W).

Step 2: Determine the constraints
The farmer has 100 feet of fence available, so the total length of fence required would be equal to the perimeter of the pen.

Perimeter = 2L + W + L

Since one side of the pen is formed by the barn, we don't need to account for that side in the perimeter calculation. Therefore, the equation becomes:

100 = 2L + W + L

Step 3: Simplify the equation
Combine like terms to simplify the equation:

100 = 3L + W

Step 4: Express one variable in terms of the other
In order to maximize the area, we need to express one variable in terms of the other. Let's solve for W in terms of L:

W = 100 - 3L

Step 5: Express the area of the rectangle
The area (A) of a rectangle is given by multiplying the length and width:

A = L * W

Substitute the expression for W from step 4 into the area equation:

A = L * (100 - 3L)

Step 6: Optimize the area
To find the largest possible area, we need to maximize the area function. We can do this by finding the critical points, where the derivative of the area function with respect to L equals zero.

Differentiate the area function:

dA/dL = 100 - 6L

Set the derivative equal to zero and solve for L:

100 - 6L = 0
6L = 100
L = 100/6
L = 16.67 (rounded to two decimal places)

Step 7: Calculate the width
Substitute the value of L back into the expression for W:

W = 100 - 3L
W = 100 - 3(16.67)
W = 50 (rounded to two decimal places)

Step 8: Calculate the maximum area
Substitute the values of L and W into the area equation:

A = L * W
A = 16.67 * 50
A = 833.5 square feet

Therefore, the largest area the farmer can enclose with 100 feet of fence and using part of his barn is approximately 833.5 square feet.