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!!