A rectangular livestock pen with THREE SIDES of fencing is to be built against the barn. The fencing is 1050ft long. Find the dimensions of the maximum area that can be enclosed. What is the maximum area?

Let the length parallel to the barn by y

let the other two sides be x each

2x + y = 1050
y = 1050-2x

area = xy
= x(1050-2x)
= -2x^2 + 1050x

quickest way: Calculus
d(area)/dx = -4x + 1050
=0 for max area
x = 262.5 ft
y = 525
the max area is 525(262.5) or 137812.5 ft^2
when the field is 525 ft long and 265.5 ft wide

or

find the vertex of the matching parabola
the x of the vertex is -b/(2a) = -1050/-4 = 262.5
sub that into
y = 1050-2x to get y = 525
so area = -2(x-262.5)^2 + 137812.5

then maximum area = 137812.5

or you could complete the square and find the vertex that way

To find the dimensions of the maximum area that can be enclosed by the rectangular livestock pen, we need to determine the length and width of the pen.

Let's assume the length of the pen is 'x' feet and the width is 'y' feet. Since the pen is rectangular, we have two lengths and two widths forming three sides of the fencing. The fourth side is the barn.

The total length of the three sides of fencing is 1050ft. This means that the sum of the lengths of the two widths and the one length is 1050ft. Therefore, we can express this relationship as an equation:

2x + y = 1050 (Equation 1)

Next, we want to find the maximum area that can be enclosed. The area of a rectangle is given by the formula A = length × width.

In this case, the area is A = xy. We want to maximize this area.

To find the maximum area, we can use the previously derived equation (Equation 1) to express either 'x' or 'y' in terms of the other variable. Then substitute this expression into the area formula to get a function of a single variable. We can then find the maximum value of this function.

From Equation 1, we can express 'y' in terms of 'x':

y = 1050 - 2x

Substituting this expression for 'y' into the area formula, we get:

A = x(1050 - 2x)

Now, we have the area as a function of 'x'. To find the maximum area, we can take the derivative of this function with respect to 'x', set it equal to zero, and solve for 'x'. This will give us the value of 'x' that maximizes the area.

Differentiating A = x(1050 - 2x) with respect to 'x', we get:

dA/dx = 1050 - 4x

Setting dA/dx equal to zero, we have:

1050 - 4x = 0

Solving this equation for 'x', we find:

x = 1050/4 = 262.5

Now, we can substitute this value of 'x' back into Equation 1 to find the value of 'y':

2x + y = 1050
2(262.5) + y = 1050
y = 1050 - 525
y = 525

Therefore, the dimensions of the maximum area that can be enclosed are x = 262.5ft and y = 525ft.

To find the maximum area, substitute these values into the area formula:

A = xy
A = (262.5)(525)
A = 137,812.5 square feet

So, the maximum area that can be enclosed by the rectangular livestock pen is 137,812.5 square feet.