Optimization problem, solve for ( ) Some years from now you are working for a book publisher. He asks you to give him a formula that will tell him the length and width of a book page that contains A square inches of printed text, a left margin of L inches, a right margin of R inches, a top margin of T inches, and a bottom margin of B inches, and that otherwise has an area as small as possible. After dusting off your Calculus notes you tell him that the length of that page equals (          ) inches, and the width equals (            ) inches. Of course, both of your answers are in terms A, L, R, T, and B. A week later your boss tells you that unfortunately, now that he has that formula, he no longer requires your services. Things turn out well, however, since with your newly refreshed Calculus skills you land a job at an engineering company that pays you twice your old salary.
I do not understand how I could express this answer without including a X or Y which represent the vertical and horizontal length of the paper.
 A: This will follow the labeling described in Tony Piccolo's comment.  You are correct that you will have to "insert" variables for the dimensions of the page itself, in order to produce a function to be optimized.  (Alternatively, one could introduce variables for the dimensions of the printed area on the page.)  You will find at times descriptions of some problem statements for which you will need to choose variables to represent unknown quantities, which are implicit in the problem, but not specified.

If we decide to call the width of the sheets of paper to be used for the book pages $ \ X \ $ and the height of the sheets  $ \ Y \ $ , then the area of the region within which text is printed will be as he states,
$$ A \ = \ ( \ X \ - \ [L + R] \ ) \ ( \ Y \ - \ [T + B] \ ) \ \ . $$
We seek to minimize the area of the sheets used, which is $ \ XY \ $ .  (Since the sizes of all four margins are fixed, this will have an effect on the dimensions of the text region.)
For single-variable calculus, we will need to "eliminate" one of the variables.  If we choose to replace $ \ Y \ $ , we must solve for it, obtaining
$$ Y \ = \ (T + B) \ + \ \frac{A}{X \ - \ (L + R)} \ \ . $$
The area function we wish to optimize is then
$$ XY \ = \ (T + B)X \ + \ A \left[\frac{X}{X \ - \ (L + R)} \right] \ \ . $$
Differentiating with respect to $ \ X \ $ and setting the expression equal to zero produces
$$ (T + B) \ + \ A \left[  \frac{(X \ - \ L \ - \  R) \ - \ X}{[ \ X \ - \ (L + R) \ ]^2} \right] \ = \ (T + B) \ - \ A \left[  \frac{L \ + \ R}{[ \ X \ - \ (L + R) \ ]^2} \right] \ = \ 0 $$
$$ \Rightarrow \ \ [ \ X \ - \ (L + R) \ ]^2 \ = \ A \left(  \frac{L  +  R}{T  +   B} \right) $$
$$ \Rightarrow \ \ X  \ = \ (L + R) \ + \ \sqrt{A \left(  \frac{L  +  R}{T  +   B} \right) } \ \ , $$
$$ Y \ = \ (T + B) \ + \ \frac{A}{\left[ \ (L + R) \ + \ \sqrt{A \left(  \frac{L  +  R}{T  +   B} \right) } \ \right] \ - \ (L + R)} $$
$$ = \ (T + B) \ + \ \sqrt{A \left(  \frac{T  +   B}{L  +  R} \right) } \ \ . $$
The printed region then has a width $ \ \sqrt{A \left(  \frac{L  +  R}{T  +   B} \right) } \ $ and a height $ \ \sqrt{A \left(  \frac{T  +   B}{L  +  R} \right) } \ $ , which we can see has the desired area.
Note that the total "horizontal" and total "vertical" margins always appear as constants in the calculations; the minimization only depends on the proportions of the margin totals, and not on how the text "block" is positioned on the page.
To illustrate a specific result, we will choose to have a printed region of area  $ \ A \ = \ 150 \ \text{cm}^2 \ $ , and will choose $ \ T \ = \ B \ = \ R \ $ , with the left margin on right-hand pages to be $ \ L \ = \ 1.5 \ R \ $ .  The text block then has dimensions
$$ \text{width:} \quad \sqrt{150 \left(  \frac{1.5R  +  R}{R  +   R} \right)} \ = \ \sqrt{150 \ \cdot 1.25} \ \approx \ 13.7 \ \text{cm} \ \ ,  $$
$$ \text{height:} \quad \sqrt{150 \left(  \frac{R  +   R}{1.5R  +  R} \right)} \ = \ \sqrt{\frac{150 }{1.25}} \ \approx \ 11.0 \ \text{cm} \ \ .  $$
We still need to choose a specific measurement in order to find the dimensions of the page sheets, but this does not alter the fact that the sheet is of minimal area for that text block.  If we set $ \ R \ = \ 2 \ \text{cm} \ $ , then  $ \ X \ \approx \ (3 + 2) \ + \ 13.7 \ \approx \ 18.7 \ \text{cm} \ $ and $ \ Y \ \approx \ (2 + 2) \ + \ 11.0 \ \approx \ 15.0 \ \text{cm} \ $ .  The pages have proportions pretty close to  5:4  , which is credible for the dimensions of a printed book.  This indicates that book dimensions are likely chosen generally to make best use of the paper.
[One point we ought to check is that we have actually found the minimal area. The second derivative for the function $ \ XY \ $ is 
$$ \frac{d}{dX} \ \left( \ (T + B) \ - \ A \left[  \frac{L \ + \ R}{[ \ X \ - \ (L + R) \ ]^2} \right] \ \right) \ = \    \frac{A \ [L \ + \ R]}{[ \ X \ - \ (L + R) \ ]^3}  \ \ ;  $$
since all of the constants are positive measurements, and the denominator is positive as well, this second derivative is positive, indicating that we have indeed found the minimal area for the page sheets.]
