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A rectangular page is to have a printed area of 62 square inches. If the border is to be 1 inch wide on top and bottom and only 1/2 inch wide on each side find the dimensions of the page that will use the least amount of paper

Can someone explain how to do this?

I started with:

$A = (x + 2)(y + 1) $

Then I isolate y and come up with my new equation:

$A = (x+2)\left(\frac{62}{x + 2}{-1}\right)$

Then I think my next step is to create my derivative, but wouldn't it come out to -1?

Anyways, I would appreciate if someone could give me a nudge in the right direction.

EDIT

How does this look for a derivative?

$A = \left(\frac{x^2-124}{x^2}\right)$

Then to solve: $ {x} = 11.1 $

$ y = 98 / 11.1 $

Does that seem about right?

If not, the only thing I would have left is setting it to 0 and solving.

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    $x\ne 11.1$, though it is close. I would leave it as $\sqrt{124}$ Then $y=62/x$ and the fact that$62$divides 124 should give you heart that you have it right. Your final answer should be the dimensions of the paper, not $x, y$. This goes back to André Nicolas second comment: *write down* what$x$and$y$are so you and we remember.2012-05-03

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Hint: How did you get the term $\left(\frac {98}{x+2}-1\right)$? You should have $62=xy$ to give the desired printable area, so $A=(x+2)(\frac{62}x+1)$. Then, you are right, you should take $\frac {dA}{dx}$ and set it to $0$ to find $x$.

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    Your response was first and lead me in the right direction. Than$k$ you!2012-05-03
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If the dimensions of the printed area are $x$ and $y$, where $y$ is the dimension with the $1/2$ inch borders (the "width"), then the printed area is $\tag{1}62= x y.$ You want to minimize the area of the entire page, which is $\tag{2}A=(x+2)(y+1).$ We want $A$ expressed in terms of one variable only; so solve $(1)$ for $y$ $\tag{3} y={62\over x } $ and substitute into $(2)$, giving $\tag{4} A(x) = (x+2)\cdot\textstyle\bigl( {62\over x }+1\bigr) . $ Now you want to minimize $A(x)$ over $x\in(0,\infty)$. Do this using the normal derivative analysis (remember to examine what happens when $x$ is close to $0$ and when $x$ is big).

Once you've found the value of $x$ that minimizes $(4)$, remember to state the answer to the question explicitly; for example "the dimensions of the paper are $x+2$ inches top to bottom and $y+1$ inches wide" (you can use $(3)$ to find the value of $y$ once you have $x$.

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    Thank you for the reply. Once I register, I'll be sure to come back and upvote.2012-05-03