A printed page is to have a total area of 80 sq. cm with a margin of 1 cm at the top and on each side and a margin of 1.5 cm at the bottom. What should be the dimensions of the page so that the printed area will be maximum?

8 cm × 10 cm

The correct answer is Option (2) → 8 cm × 10 cm

Let $x\, cm (x > 0)$ be one dimension of the page, then the other dimension is $\frac{80}{x}cm$, for the area of the page is given to be 80 sq. cm.

Let A $cm^2$ be the printed area, then

$A=(x-2)\left(\frac{80}{x}-2.5\right)=85-\frac{5}{2}x-\frac{160}{x}$.

Diff. w.r.t. x, we get

$\frac{dA}{dx}=-\frac{5}{2}x+\frac{160}{x^2}$ and $\frac{d^2A}{dx^2}=-\frac{320}{x^3}$.

Now $\frac{dA}{dx}=0⇒-\frac{5}{2}+\frac{160}{x^2}=0⇒x^2=64$ but $x>0$

$⇒x=8$

Also $\left(\frac{d^2A}{dx^2}\right)_{x=8}=-\frac{320}{512}=-\frac{5}{8}<0$ ⇒ A is maximum at $x = 8$.

∴ The dimensions of the page are 8 cm and $\frac{80}{8}$ cm i.e. 10 cm.