Question

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?

(Answer should be small and large)

Answer 1

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.

Explanation:

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.!