# Math Help - [SOLVED] Optimization Problem

1. ## [SOLVED] Optimization Problem

Hi,

I just need a clue to what to do next for this question:
Each rectangular page of a book must contain 43.890625 cm$^2$ of printed text, and each page must have 2.5 cm margins at top and bottom, and 1.6 cm margin at each side. What is the minimum possible area of such a page?

Solution: Let $x$ be the width and let $y$ be the height of the page. Then the width of the printed area is given by $l =$ x-3.2 and the height of the printed area is given by $h$ = y-5 . We note that the area of the printed text is given by $p$ = (x-3.2)(y-5) . The problem is to find $x$ and $y$ to minimize xy under the condition p= (x-3.2)(y-5) = 43.890625.

The problem I am stuck with, is finding the area in terms of x...
We aren't given another equation to isolate y, and I tried isolating for y in terms of the area which gave me
43.890625 + 5x - 16 = y
but the only possible answers all contained 43.890625 without being maniulated...

2. Originally Posted by alleysan
Hi,

I just need a clue to what to do next for this question:
Each rectangular page of a book must contain 43.890625 cm$^2$ of printed text, and each page must have 2.5 cm margins at top and bottom, and 1.6 cm margin at each side. What is the minimum possible area of such a page?

Solution: Let $x$ be the width and let $y$ be the height of the page. Then the width of the printed area is given by $l =$ x-3.2 and the height of the printed area is given by $h$ = y-5 . We note that the area of the printed text is given by $p$ = (x-3.2)(y-5) . The problem is to find $x$ and $y$ to minimize xy under the condition p= (x-3.2)(y-5) = 43.890625.

The problem I am stuck with, is finding the area in terms of x...
We aren't given another equation to isolate y, and I tried isolating for y in terms of the area which gave me
43.890625 + 5x - 16 = y
but the only possible answers all contained 43.890625 without being maniulated...

the printed area is fixed ... the problem wants to minimize the area of the entire page.

let x = width of printed part
y = length of printed part

$A = (x+3.2)(y+5)$

$xy = 43.890625$ cm$^2$ ... call that ugly area constant, $k$

$y = \frac{k}{x}
$

$A = (x + 3.2)\left(\frac{k}{x} + 5\right)$

find $\frac{dA}{dx}$ and optimize