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.
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
Area = xy
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.
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!!