A parking lot is to be formed by fencing in a rectangular plot of land except for a n entrance 12m wide along one of the sides. Find the dimensions of the lot of the greatest area if 300m of fencing is to be used.

### 1 Answer

Let parking lot be of dimension x by y. Say that the 12m wide unfenced section is in one of the x sections:

Add up the lengths of all sections to 300m:

x + (x-12) + y + y = 300

2x + 2y = 312

x + y = 156

x = 156 – y

And:

Area = xy

Combine equations:

Area = (156 – y)y

Area = -y^2 + 156y

To maximize area, you should have been taught to set the variable to -b/2a (for a quadratic equation ax^2 + bx + c)

In this case:

y = -156 / (2*-1)

y = 78 meters

x = 156 – y = 78 meters

So the dimensions of the lot are 78 meters by 78 meters.

Error check:

The four fence sections should add up to 300 meters. Remember, one section is reduced by 12 meters:

78 + 78 + 78 + (78-12) = 300. Checks!!