<p>A farmer has 1500ft of fence. He plans to fence a rectangle of land next to a river. (such that the fence is not along the river so it only has 3 sides) What is the largest area he can enclose?</p>
<p>Are you sure it’s 1500 ft of fence? because if it is, the answer (140625) wouldn’t fit for a SAT Math Grid-in question…</p>
<p>but to maximize your area for any rectangle, you would make your rectangle a square. You could simply divide 1500 by 4 and square that answer, or you could use algebra to find your answer.</p>
<p>A=x*y
P=1500=2x+2y</p>
<p>2y=1500-2x<br>
y=750-x</p>
<p>A=x*(750-x)=750x-x^2=-x^2+750x</p>
<p>The x-coordinate of the vertex is given by -b/2a in a quadratic equation ax^2 + bx + c.
x-coordinate of the vertex= -750/(2*-1)=375</p>
<p>750-375=375, so your rectangle is a square.
You could also graph this in your graphing calculator to check really fast.</p>
<p>Edit: So, Josh66, is this from the ACT or SAT? I can’t tell even though I’ve taken both tests…</p>
<p>It was actually a problem my friend had on his pre-calc midterm, I realize now this probably wasn’t the best section to put it in. anyway, it only had 3 sides so it would have been y=1500-2x, but thanks so much for the help</p>
<p>Lol. Why would a rectangle only have 3 sides though?</p>
<p>It borders a river so it doesn’t have to be fenced on one side lol</p>
<p>Ohhhh. Nice one! The place where I live doesn’t have many rivers… the one I (seldomly) see is right by a chemical plant, so it’s not very nice. Got any nice rivers where you live? 
So its maximum area would be 250k, right? If it only had 3 sides?</p>
<p>No you had the right process the first time lol
A=x*y
P=1500=2x+y</p>
<p>y=1500-2x </p>
<p>A=x*(1500-2x)=1500x-x^2=-2x^2+1500x</p>
<p>-b/2a = -1500/-4 = 375</p>
<p>so A=xy= (375)(1500-2[375])=(375)(750)= 281250ft</p>
<p>And what exactly is the reasoning for using -b/2a? I understand its right but when are you allowed to use that in a problem such as this one?</p>
<p>Mmmm. Ok, I see. xD</p>
<p>Don’t you remember “-b/2a” from your Pre-Calc class? I understand that it’s the x-coordinate of the vertex finder from looking at the Quadratic Formula.</p>
<p>x=-b +/- sqrt (b^2- 4ac)
---------------------
2a</p>
<p>You might not remember since it might’ve been a while since your last Alg 2 or Pre-Calc class. What math are you in again?</p>
<p>I don’t really know how to explain how I know that the vertex is -b/2a… I think I knew it intuitively or my math teacher told us to do that… but here’s a technical explanation online:
[The</a> Vertex of a Parabola](<a href=“http://hotmath.com/hotmath_help/topics/vertex-of-a-parabola.html]The”>The Vertex of a Parabola) .</p>