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?

Answer 1

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