Iwould like to create a rectangular vegetable patch. the fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. i have a budget of $128 for the project. what are the dimensions of the vegetable patch with the largest area i can enclose?

north and south sides ?
east and west sides

The answer to this question:
north and south sides 16ft
east and west sides 8ft

In this question, the east and west fencing cost is $4/ft but the south and north fencing cost is $2/ft. The cost is cheaper for fencing north and south side which cost half of the east and west side. The perimeter is made by adding the south, north, east and west side.
If x = north and south fencing and y= is east and west side, then you can get this equation

cost= (south+north) * $2 + (east+west)*$4  
128=(x+x) * 2 + (y+ y) * 4
128= 4x + 8y
32= x+ 2y
x= 32-2y

The area is count by multiplying the width and length, which was x and y in this question. The formula should be:
area= length * width
area= x * y
area= (32-2y) * y
area= 32y - 2y^2

Then, you need to find the value of y needed for the maximum area.
32y- 2y^2 
32 - 2*2y=0
32= 4y
y= 8

If y=8 then
x= 32-2y
x= 32- 16= 16

Ireally don’t understand your problem do you have a better picture of it