A farmer wants to fence in a rectangular pen using the wall of a barn for one side of the pen and 115 feet of fencing for the remaining 3 sides. What dimensions will give her the maximum area for the pen?

if the pen has dimensions x by y, with x along the barn, then

x+2y = 115

the area a is thus

a = xy = (115-2y)y

this is a parabola, with maximum at y=115/4

so, x = 115/2

so, a pen 115/4 by 115/2 gives the maximum area

To find the dimensions that will give the farmer the maximum area for the pen, we can use the concept of calculus to optimize the problem.

Let's assume the length of the pen, parallel to the barn, is "x" feet. Therefore, the width of the pen, perpendicular to the barn, will be "y" feet.

Since one side of the pen is the wall of the barn, the total fencing required will be the sum of the three remaining sides, which is equal to 115 feet.

We can set up an equation using these variables:

x + y + x + y = 115 [The two sides parallel to the barn are equal to "x" and the side perpendicular to the barn is equal to "y"]

Simplifying the equation, we get:

2x + 2y = 115

Now, we need to express the area of the rectangular pen in terms of "x" and "y". The area, A, of a rectangle is given by:

A = length × width
A = x × y

Now we have an area equation in terms of x and y, and a constraint equation relating x and y. To find the dimensions that maximize the area, we need to find the critical points of the area function.

To do that, we can solve the constraint equation for one variable in terms of the other. Rearranging the constraint equation, we get:

2y = 115 - 2x
y = (115 - 2x)/2
y = 57.5 - x

Substituting this value of y into the area equation, we have:

A = x × (57.5 - x)
A = 57.5x - x^2

Now we have the area in terms of a single variable, x.

To find the maximum area, we need to find the critical point by taking the derivative of the area function with respect to x and setting it equal to zero:

dA/dx = 57.5 - 2x
0 = 57.5 - 2x

Solving this equation, we get:

2x = 57.5
x = 28.75

Substituting this value of x back into the constraint equation, we can find the value of y:

2y = 115 - 2(28.75)
2y = 115 - 57.5
2y = 57.5
y = 28.75

Therefore, the dimensions that will give the farmer the maximum area for the pen are x = 28.75 feet and y = 28.75 feet.