A parking lot is to be formed by fencing in a rectangular plot of land except for an entrance 12 m wide. How do you find the dimensions of the lot of greatest area if 300 m of fencing is to be used?

1 Answer
Jan 8, 2016

Answer:

The maximum area is obtained if the plot is #78#m square.

Explanation:

The perimeter of the plot must be #(300 +12)# if #300# m fencing are used and there is a #12# m entrance.
Sketch
The perimeter is given by #P=2(L+W)# and the area is given by #A=L*W#
#P=2(L+W) = 312#
#:.W = 312/2 - L = 156 - L#
#:.A = L*(156-L) =156L - L^2#

For a quadratic of the form #y = ax^2+bx+c# the vertex )which gives the maximum or minimum value) is found at the point #x=c - b^2/(4a)#
For this parking lot #c=0, b = 156# and #a=-1#
#:.A_v = -156^2/(-4) =24336/4 = 6084#

So the maximum area of the plot is 6084 sq .m.
#6084 = -L^2 +156L#
#0 = L^2 -156L +6084#
#0 = (L-78)^2#

#:. L=78#
#W=156 - l = 156 - 78 = 78#

Therefore the are of the plot is maximized if the plot is #78m# square.