Expert: Ahmed Salami - 10/4/2004

Question
Stacy has 30 yards of fencing that she wishes to use to enclose a rectangular garden. If all of the fencing is used, what is the maximum area of the garden that can be enclosed? I understand how my book got the area -x^2+15x, but then they do -b/2a to find x,which I do not understand why. Also, how do I know which situation I should use that formula for and what situation I should not?Then, to finish off the problem, they just plug it into the equation to get the area to be 56.25yd^2.And that I understand.Please explain that part about -b/2a. Thank you so much.

Answer
Hi jeff,
Thanks for making the task easier. Alright then, we'll start from where you're stuck.
Generally, for a quadratic equation of the form
ax^2 + bx + c, the minimum or maximum value occurs at the point when x = -b/2a. We could look at it this way, observe the graph of any quadratic equation and you would see that this point surely lies exactly in the middle of the roots. I'm sure that you know the formula for the roots of quadratic equations, add both the roots and divide by 2, you'll get the value -b/2a.
The other way around it is calculus and i'm assuming you've done a bit of it.
If y = ax^2 + bx + c
dy/dx = 2ax + b
and maximum or minimum points occur when dy/dx = 0, i.e when
2ax + b = 0, which gives us
x = -b/2a
I'm sure you now understand it, if not you know what to do, just get back to me.
Good luck.
Regards.