What width will produce the maximum garden area?

Question

A(w)=−w(w−80)

Answer 1

Hi Nevaeh,

There are two ways we can think about doing this: analytically or graphically.

Let's start with the analytic approach:

The given function forms a parabola, it might be easier to see this if we expand it first.

A(w) = -w² + 80w

Notice that the squared term (-w²) is negative, which means that the parabola will be facing down like a frowny face. This tells us that the vertex will give us the highest point on the graph (aka the maximum). Translating this into terms of the question, we need to find the ordered pair (x, y) or (w, A) of the vertex of our graph. If our quadratic is in the form ax² + bx + c, we can find this point by using the formula for the vertex :

(-b/2a, f(-b/2a))

We have a = -1, and b = 80... so plugging in for the w-coordinate of our point we get -b/2a = -80/2*-1 = 40. This tells us that the maximum area of the garden will be produced by having a width of w = 40.

Alternatively, we could take a more graphical approach to this. Go ahead and graph this function using a graphing calculator or website. If you find the maximum, you will notice it is at the point (40, 1600). This is an easier way of doing it, but it requires that you have access to technology.

Hope this helps!