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.!