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?

Question

(Answer should be small and large)

19915 views around the world

You can reuse this answer
Creative Commons License

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.!