Answer to Question #343546 in Calculus for yousii

Question

A poster is to have an area of 630 cm2 with 2.5 cm margins at the bottom and sides and a 5 cm margin at the top. Find the exact dimensions (in cm) that will give the largest printed area.

width : cm

height: cm

Answer 1

Let $x=$ width of the poster, let $y=$ height of the poster.

Then

$$xy=630$$

The largest printed area will be

$$A=(x-2.5-2.5)(y-2.5-5)$$

Substitute

$$A=A(x)=(x-5)(\dfrac{630}{x}-7.5)$$

$$=630-7.5x-\dfrac{3150}{x}+37.5$$

$$=667.5-7.5x-\dfrac{3150}{x}, 5\le x\le 84$$

Differentiate wih respect to $x$

$$A'=-7.5+\dfrac{3150}{x^2}$$

Find the critical number(s)

$$A'=0=>-7.5+\dfrac{3150}{x^2}=0$$

$$x=\pm\sqrt{420}$$

Since $5\le x\le 84,$ we take $x=\sqrt{420}$

If $5\le x<\sqrt{420}, A'>0, A$ increases.

If $\sqrt{420}<x\le\sqrt{420}, A'<0, A$ decreases.

The function $A$ has the absolute maximum for $5\le x\le 84$ at $x=\sqrt{420}.$

$$y=\dfrac{630}{\sqrt{420}}=\sqrt{945}$$

width : $\sqrt{420}$ cm

height: $\sqrt{945}$ cm