A word problem from Khan academy reads like this:
Marquise has $200$ meters of fencing to build a rectangular garden.
The garden's area (in square meters) as a function of the garden's width $w$ (in meters) is modeled by:
$A(w) = −w^2 +100w$
What is the maximum area possible?
Since the problem talks about maximum and the equation is a quadratic, I factored the thing to vertex form:
$$y = -1(w – 50)^2 + 2500$$
And the solution they are looking for is indeed $A(50) = 2500$ (square meters).
But I don't really understand what is modeled here. What does it mean to have the area of a rectangle being modeled by a function of its width? Somehow I'm stumped by this example.
Best Answer
Your function tells you the area of your rectangular garden if you use all 200 meters of fencing and it has width w.
This function can be obtained by manipulating the relationship between length, width, the perimeter, and the area.
Let $l$ be the length and $w$ be the width. Then $$2(l+w)=200,$$ since your perimeter is the sum of the lengths of all four sides, and $$lw=Area.$$
Now, we manipulate the perimeter equation to get \begin{align}l+w&=\frac{200}{2}\\ l&=200/2-w\\ &=100-w. \end{align}
Substituting this into the area expression gives $$Area=(100-w)w=100w-w^2.$$