# 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)

Apr 12, 2017

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

$\text{Length=} \left(l + 2\right) = 6$ inch, and, the $\text{Width=} \left(\frac{16}{l} + 2\right) = 6$ 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 $\frac{16}{l}$

inch.

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

length of the page must be $\left(l + 2\right)$ inch, and, smilarly, the width, 

(16/l+2) inch.

These give us, the Area of the page $\left(l + 2\right) \left(\frac{16}{l} + 2\right) = 16 + 2 l + \frac{32}{l} + 4 ,$

$\mathmr{and} , 20 + 2 \left(l + \frac{16}{l}\right) ,$ which, being a fun. of $l ,$ we write, it as,

$A \left(l\right) = 20 + 2 \left(l + \frac{16}{l}\right) \ldots \ldots \ldots \left(1\right)$

To find the least amt. of paper, we need to minimise $A \left(l\right) .$

Knowing that, for ${A}_{\min} , A ' \left(l\right) = 0 , \mathmr{and} , A ' ' \left(l\right) > 0.$

$\text{From } \left(1\right) , A ' \left(l\right) = 0 \therefore 2 \left\{1 - \frac{16}{l} ^ 2\right\} = 0 \therefore {l}^{2} = 16 \therefore l = \pm 4$.

$A ' ' \left(l\right) = 2 \left\{0 - 16 \cdot \left(- 2\right) {l}^{-} 3\right\} = \frac{64}{l} ^ 3 \Rightarrow A ' ' \left(+ 4\right) = 1 > 0.$

$\therefore l = + 4 \text{ gives } {A}_{\min} .$

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

$\text{Length=} \left(l + 2\right) = 6$ inch, and, the $\text{Width=} \left(\frac{16}{l} + 2\right) = 6$ inch.

Enjoy Maths.!