# Homework Help: Introductory Calculus help

1. May 14, 2014

### Fifty

"The printed area of a book will be 81 cm2. The margins at the top and bottom of the page will each be 3 cm deep. The margins at the sides of the page will each be 2 cm wide. What page dimensions will minimize the amount of paper?" (Calculus and Vectors, Peter Crippin et al., 158).

[I added the citation just in case!]

I first thought...
"the width of the entire page will be x and the length will be y. The width of the printed area then is x - 4 (excluding the side margins) and the length of the printed area is y - 6. So (x - 4)(y - 6) = 81 cm2.

The total area is A = xy. y, in terms of x, is y = $\frac{1000}{x - 4}$ + 6. So then I figured:

A(x) = x ( $\frac{1000}{x - 4}$ + 6 ) would be the area and I would take its derivative and set A'(x) = 0."

I always end up with 81x - 81x in the differentiated function and get 0 = -324. I've tried multiple approaches (including setting x to the width of the printed page and then the total width being x + 4) and using different methods to get to the derivative, but this always happens.

Thanks for the help!

EDIT: Whoops! Thanks, chet!

unsimplified:
A'(x) =$\frac{81(x-4) - 81x}{(x - 4)}$

Sorry, apparently formatting the fraction won't work if I include the exponent formatting as well, but the (x - 4) in the denominator is supposed to be squared.

Last edited: May 14, 2014
2. May 14, 2014

### Staff: Mentor

It's hard to help unless you show us what you got for the derivative.

Chet

3. May 15, 2014

### pasmith

If $(x - 4)(y - 6) = 81$ then surely $y = \frac{81}{x - 4} + 6$?

(LaTeX fixed)

I think you're missing a $6(x-4)^2$ from the numerator; you should have $A'(x) = xy'(x) + y(x)$.

4. May 15, 2014

### Fifty

I see what I did wrong; I didn't differentiate correctly. I got it now, thanks :D