Please help with this simple word problem

A rectangular field is to be bounded by a fence on three sides and by a straight stream on the fourth side.

a) Find the exact area of the field with maximum area that can be enclosed using 13,457 feet of fence.

b) The side parallel to the stream is_____feet.
The two sides perpendicular to the stream are each____feet.

Please explain. TY

Let x be the length of the side parallel to the stream.
Then the length of one of the other sides is (13457 - x) / 2.
Area = x (13457 - x) / 2 = (1/2) (13457x - x^2)
Take the derivative:
13457/2 - x
Set the derivative to zero:
13457/2 - x = 0
Solve for x:
x = 13457/2

a)
Max area = (13457/2) (13457 - (13457/2)) / 2 = (181090849 / 8) square feet

b) The side parallel to the stream is 13457 / 2 feet.
The two sides perpendicular to the stream are each 13457 / 4 feet.

The maximum area using the smallest perimeter will be achieved if it has the shape of a square.


For example lets look at a rectangle with an area of 16
if the length was 16 and the width 1 the Area is 16 ft^2 and perimeter = 2*16 + 2*1 = 34 ft
if the length was 8 and the width 2 the Area is 16 ft^2 and perimeter = 2*8 + 2*2 = 20 ft
If the length was 4 and the width 4 the Area is 16 ft^2 and perimeter = 2*4 + 2*4 = 16 ft

This indicates that greatest area with the least perimeter can be contained in a square.

So you have 13,457 feet of fence that will need to make up 3 of the 4 sides.

Because you want it to be a square all sides need to be equal so
13,457 feet/3 = 4485.66666

a) The area is the side squared:
4485.66666^2 = 20121205.44 ft^2

b) Because it is a square, all sides are the same:
The side parallel to the stream is 4485.66666 feet.
The two sides perpendicular to the stream are each 4485.66666 feet.

I hope this helps!