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?

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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?

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Answer 1 · 2016-01-08T10:14:06

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 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.

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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

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