Question

The printed text takes $432c{m}^2$ of a page. The width of a margin at the top and bottom of the page is 2 cm and that of the lateral margins is 1.5 cm. What must be the width and the height of the page for the quantity of the paper used to be the least?

Answer 1

Total area of the paper $=lb$

$l\to height$

$b\to width$

$[b-2(1.5\left)\right][l-2(2\left)\right]=432c{m}^2$

$b-3=\frac{432}{l-4}$

$b=3+\frac{432}{l-4}$

$Area=lb=l(3+\frac{432}{l-4})$

For area to be minimum

$0=\frac{d\left(Area\right)}{dl}=l\left(\frac{432}{(l-4{)}^2}\right)+3+\frac{432}{l-4}$

$=\frac{32l}{(l-4{)}^2}+3+\frac{(l-4)432}{(l-4)2}$

$0=3-\frac{432l}{(l-4{)}^2}\Rightarrow 576=(l-4{)}^2$

$l-4=\pm 24$

$l=28cm$

$\therefore b=21cm$