Algebra 1 Review Quadratics Word Problem

[MasonD]

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.

 

[MarkFL]

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:

[MATH]A=LW[/MATH]
Now, suppose the gate is put on one of the sides of length \(W\). We then have:

[MATH]300=2L+W+W-12=2(L+W)-12[/MATH]
Hence:

[MATH]312=2(L+W)[/MATH]
[MATH]156=L+W[/MATH]
And now we can give the area in one variable...let's choose \(L\):

[MATH]A=L(156-L)[/MATH]
Does that make sense so far?

 

[MasonD]

Thank you! That helped a lot.

 

[MarkFL]

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?