I've been struggling with this for a while but cannot form an equation that will differentiate to give a minimum.
Question is as follows:
The pages of a book are to have margins of $\frac{1}{2}$ inch at the sides of the print and $1$ inch at the top and bottom. If the area of the print is to be $32$ sq. in. find the dimensions of a page of minimum area.
(HINT: let $x$ inches denote width of the print.)
Answer given: $5$ inch by $10$ inch.
[My interpretation of the question as a diagram]
Despite numerous attempts , the best I can come up with is:
$$y=\frac{32}{x}$$
which has no minimum.
My working is as follows:
$$(x+1)(y+2)-2(x+1)-y=32$$
i.e. total area of page subtract area of margins.
Which simplifies to:
$$xy=32$$
If anyone can offer any advice on this, then I would be very grateful. It may be possible, perhaps probable, that I have interpreted this question in completely the wrong way. In fact there must be a mistake in my reasoning or I would have derived the correct result.
Many thanks in advance for any advice given.
Note that the function to be minimised is not xy=32. This is the relation obtained so that the area of "print" remains 32.
What has to be minimised is the area of the "page" (that is, the area of margins + area of print).
So, make the function for page area as A:(x+1)(y+2). Now we can use xy=32 and put y as $\frac{32}{x}$ in A.
This gives A(x)=(x+1)($\frac{32}{x}$ +1). Differentiating for the minima, x=4 and y= 32/4. The page dimensions are hence, 4+1 and 8+2 = 5 and 10 inches