I need help finding the area of this corral

Question

Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing. (Assume that the length is greater than or equal to the width.)

Length in ft. =

Width in ft. =

Answer 1

Let x be the width of the rectangle and y be the length of the rectangle.

+----------+----------+

| | | x

| | |

+----------+----------+

y/2 y/2

The fence length of the rectangle for 2 pens is:

y/2 + y/2 +y/2 +y/2 + 3x = 300

or

2y+3x=300 or y = (300-3x)/2

The area of the rectangle, A = xy = 150x-3x²/2

Which is the equation for a down opening parabola. The maximum area will be at the vertex of the parabola.

Solving for the algebra 1 way

The x-value of the parabola vertex is:

x = -b/2a where a = -3 and b = 150

x = -150/(2(-3)) = 50

y = (300-150)/2 = 75

So, the width is 50 ft and the length is 75 ft

If this was calculus, take the derivative of the area, set it to zero and solve for x:

A'(x) = 150-3x

150-3x=0

x = 50