I would 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 ______?

Question

I would 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    ______?

anonymous · Answer

put length of east and west side as x  north and south side as y
then you know 
$$4\times 2x+2\times 2y=128$$
$$8x+4y=128$$
$$4y=128-8x$$
$$y=32-2x$$

anonymous · Answer

Area is 

$$A=xy$$ so $$A(x)=x(32-2x)=32x-2x^2$$

anonymous · Answer

you want the largest area, so if this is a calc class take the derivative, set = 0 and solve, or if this is not calc say you ahve a parabola that opens down, will have the maximum at the vertex which you find by 
$$-\frac{b}{2a}=\frac{32}{4}=8$$

anonymous · Answer

so x should be 8, y should be 16

riley · Answer

Wow thank you so much for explaining it like that Satellite.  I'll go back and work through it myself following your steps, so hopefully I'll end up with the same answer as yours.  Thank you again, that was very very helpful.