a rancher has 650ft of fencing he wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.
Find the function that models the total area of the four pens. (use W to represent the width of the field and write the function A in terms of W)
Find the largest area possible of the four pens.
March 25th 2008, 04:19 PM
Okay, so let W represent the width of the pens. Note that, to make 4 separate pens, he's going to need to do a couple things:
Surround the whole rectangular area with fence AND
Put up 3 "extra" fences inside, parallel to the width, each with length W, to divide it into 4 pieces.
So, altogether, he's going to need 5W feet of fence to construct the widths (the two end pieces and then the three pieces inside the area). This leaves 650 - 5W feet with which to construct the two lengths. So, each length with be half that, or 325 - 2.5W.
The total area is length x width = (325 - 2.5W) * W, which works out to a quadratic if you simplify (which you probably want to do).
To find the largest possible area, we want to determine the maximum height of the quadratic we just found. The quadratic has a negative leading coefficient (-2.5), so it opens downward, meaning that the vertex is the highest point, i.e., the maximum of the function. So we need to find the vertex of the function from above.
Remember that the x-coordinate (here, the "W-coordinate") of the vertex can be found as follows:
Plug in b and a and you're done. (Remember when determining what a and b are, write the quadratic equation in descending order, i.e., write the term with the highest degree first (the x^2 term), then the x term.)