Algebra 1 Review Quadratics Word Problem

MasonD

New member
Joined
Aug 9, 2019
Messages
4
I have a problem that reads: A parking lot is to be formed by fencing in a rectangular plot of land except for an entrance 12 feet wide. Find the dimensions of the lot of greatest area if 300 feet of fencing is to be used.

I am also supposed to provide a graph for this problem. On the answer sheet it says that the equation for the graph is y=x(156-x).
Y being area and X being width. I have no idea how to get to this answer. I understand that 156 * 2 = 312, which is 300 + 12, but I don’t understand how to get this equation. Could somebody please explain it to me? Thanks.
 
Last edited:

MarkFL

Super Moderator
Staff member
Joined
Nov 24, 2012
Messages
2,165
Hello, and welcome to FMH! :)

Let's let \(L\) be the length of the lot, and \(W\) be the width. And so the area \(A\) is:

\(\displaystyle A=LW\)

Now, suppose the gate is put on one of the sides of length \(W\). We then have:

\(\displaystyle 300=2L+W+W-12=2(L+W)-12\)

Hence:

\(\displaystyle 312=2(L+W)\)

\(\displaystyle 156=L+W\)

And now we can give the area in one variable...let's choose \(L\):

\(\displaystyle A=L(156-L)\)

Does that make sense so far?
 

MasonD

New member
Joined
Aug 9, 2019
Messages
4
Thank you! That helped a lot.
 

MarkFL

Super Moderator
Staff member
Joined
Nov 24, 2012
Messages
2,165
Thank you! That helped a lot.
Okay, and so I would continue with the fact that the vertex of a parabola, where the max/min occurs, lies along the axis of symmetry, which will be midway between the roots. We have the area function already factored, so where would the axis of symmetry be?
 
Top