A rectangular page is to contain 16 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used?

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Answer 1 · 2017-04-12T18:50:56

For the least usage of paper, the reqd. dimns. of the page, are,

#"Length="(l+2)=6# inch, and, the #"Width="(16/l+2)=6# inch.

Let #l# inch be the length of printed rectangular region of the page.

Since, the Area of the printed rectangular region has to be #16#

sq.in., we find that, the width of the prited portion must be #16/l#

inch.

Now, the margin of #1# inch has been left on both sides, so, the

length of the page must be #(l+2)# inch, and, smilarly, the width, #

(16/l+2)# inch.

These give us, the Area of the page #(l+2)(16/l+2)=16+2l+32/l+4,#

#or, 20+2(l+16/l),# which, being a fun. of #l,# we write, it as,

#A(l)=20+2(l+16/l).........(1)#

To find the least amt. of paper, we need to minimise #A(l).#

Knowing that, for #A_(min), A'(l)=0, and, A''(l) >0.#

#"From "(1), A'(l)=0:. 2{1-16/l^2}=0:.l^2=16:.l=+-4#.

#A''(l)=2{0-16*(-2)l^-3}=64/l^3 rArr A''(+4)=1 >0.#

#:. l=+4" gives "A_(min).#

Thus, for the least usage of paper, the reqd. dimns. of the page, are,

#"Length="(l+2)=6# inch, and, the #"Width="(16/l+2)=6# inch.

Enjoy Maths.!