finding dimensions

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 $144 for the project. What are the dimensions of the vegetable patch with the largest area I can enclose?

north and south sides____  ft

east and west sides____  ft

Answer 1

Let x be the length of the north/south and y be the length of the east/west

The total cost of north/south is 2*2*x (two sides * x feet per side * $2 per foot)

The total cost of east.west is 2*4*y (two sides * y feet per side * $4 per foot)

So the total cost is 4x+8y, which must be 144 so 4x+8y=144, or x+2y=36

You want to maximize the area, which is A=xy

The budget constraint allows you to write A in terms on one variable, instead of two (x and y)

The budget, solved for x, is x=36-2y

Substitute this is to A=xy so A=(36-2y)y or A=36y-2y^2

Find the maximum of this using calculus or the vertex of a parabola:

this gives y=9 and substituting in the constraint gives x=18

You can use a graph or calculus to confirm that this is the maximum area and not the minimum!