# Algebra 1 Review Quadratics Word Problem

#### MasonD

##### New member
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
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
Thank you! That helped a lot.

#### MarkFL

##### Super Moderator
Staff member
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?