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?

Answer 1

Let the lengths of the east and west sides be x and the lengths of the north and south sides be y.  the dimensions you want are therefore x and y.

The cost of the east and west fencing is $4*2*x; the cost of the north and south fencing is $2*2*y.  We have to put in that "2" because there are 2 sides that run from east to west and 2 sides that run from north to south.



The total cost of all this fencing is $4(2)(x) + $2(2)(y) = $128.  Let's reduce this by dividing all three terms by 4:  2x + y = 32.

Now we are to maximize the area of the vegetable patch, subject to the constraint that 2x + y = 32.  The formula for area is A = L * W.  Solving 2x + y = 32 for y, we get y = -2x + 32.

We can now eliminate y.  The area of the patch is (x)(-2x+32) = A.  We want to maximize A.

If you're in algebra, find the x-coordinate of the vertex of this quadratic equation.  Remember the formula x = -b/(2a)?  Once you have calculated this x, subst. your value into the formula for y:  y= -2x + 32.

Now multiply together your x and y values to obtain the max area of the patch.


If you're in calculus, differentiate A = x(-2x+32) with respect to x and set the derivative equal to zero.  This approach should give you the same x value as before; the corresponding y value will be the same;  y=-2x+32.

Multiply x and y together.  That'll give you the maximum possible area of the garden patch.