Question

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?

Answer 1

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.

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.