1. You have 150 yards of fencing to enclose a rectangular region. One side of the rectangle does not need fencing. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

Let the dimensions be x and y. 2 of the sides measuring x and only one of the sides measuring y should be fence.

150 = 2x + y

The area of the rectangular figure,

A = xy

Substituting the y in the first equation to the second,

A = x (150 - 2x)

A = 150x - 2x²

Differentiate the equation and equate to zero,

dA/dx = 0 = 150 - 4x

The value of x is 37.5 yard and y is equal to 75 yard. The maximum area is 2812.5 yd².