Question is down there to big for up here....?

Question

A farmer has 500 feet of fencing to use to make a rectangular garden.
One side of the garden will be a barn, which requires no fencing. How
should the pen be built in order to enclose the largest amount of area
possible?
Give your answer by filling in the blanks in the following sentence:
The farmer should build the pen ______ feet away from the barn, and
________ feet wide, for an area of _______ square feet

Answer 1

The farmer should build the pen 125 feet away from the barn, and 250 feet wide for an area of 31,250 square feet.

But do you know why?

Let x be the distance from the barn. So you will have two sides of length x, and one side of length 500 - 2x (the remaining fencing).

Written as a function for area:
f(x) = x * (500 - 2x)
f(x) = -2x² + 500x

Now massage it to get it into vertex form:
f(x) = -2(x² - 250x)

Complete the square by taking half of 250 --> 125 and squaring it --> 15,625. Now subtract and add this:
f(x) = -2(x² - 250x - 15625 + 15625)

The first 3 terms become a perfect square now:
f(x) = -2(x - 125)² + -2(15625).

And the last term simplifies to 31,250:
f(x) = -2(x - 125)² + 31,250

This means you have a vertex at the point (125, 31250) and the parabola points downward, making this point a maximum.

This point is the length of fencing from the barn (x = 125) and the area (31250). The width of fencing comes from the equation before (500 - 2x) --> 250 feet.

Length = x = 125 ft.
Width = 500-2x = 250 ft.
Area = 31,250 sq. ft.

The farmer should build the pen 125 feet away from the barn, and 250 feet wide for an area of 31,250 square feet.

Now learn this, since you won't have Y!A when it appears on the test.

Answer 2

Let x = side parallel to barn, y = sides perpendicular to barn. Then x + 2y = 500, so y = (500-x)/2 = 250 - x/2.

The area is xy, so A = x(250 - x/2) = 250x - x²/2. Since that's quadratic, it's at its max halfway between its zeros:

0 = x(250 - x/2)
x = 0 or 250 - x/2 = 0, x = 500.
Halfway between 0 and 500 is 250. So when x = 250, y = 125. The pen should run 125 feet out from the barn and be 250 feet wide (parallel to the barn).