Find the length and width of a rectangle that has the given perimeter and a maximum area? Perimeter: 164 meters

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Answer 1 · 2017-04-12T15:04:49

The Reqd. Dims. of the Rectangle for Maximum Area are

$l=41 m., and, w=82-l=41 m.$

Let $l and w$ denote the length and width of the Rectangle.

Its Perimeter is $2(l+w)$ , which is given to be, $164.$

$:. l+w=164/2=82...(1).$

Now, Area $A$ of the Rectangle, is, given by, $A=lw.$

$(1) rArr A=l(82-l)=82l-l^2,$ which is a function of $l,$ so let us

write it as $A(l)=82l-l^2.......(2)$

We are reqd. to maximise $A$ .

We know that, for $A_(max), A'(l)=0, and, A''(l) <0.$

$(2) and A'(l)=0 rArr 82-2l=0 rArr l=82/2=41.$

Further, $A""(l)=-2 < 0, AA l,$ &, in particular, for $l=41," too."$

Thus, $l=41," gives "A_(max)$ .

Hence, the reqd. dims. of the rectangle for maximum area are

$l=41 m., and, w=82-l=41 m.$

Enjoy Maths.!