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# Calculus question; at 1st year degree level, but probably A-Level standard watch

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1. Please see the attached diagram.

An editor of a publishing company decided that the pages of a book should have margins of 2.5 centimetres at the top and bottom of the page and 1.5 centimetres on each side, as in the diagram.
She further stipulated that each page should have an area of 600 square centimetres.

The questions which follow are:

(a) Determine a formula for the printed area of the page A in terms of h or W but not both.

(b) Find the page dimensions that will result in the maximum (inner) printed area on the page and state this area.

(c) Plot a suitably labelled graph and interpret it to show how it verifies your findings.

The slightest bit of help will be appreciated, thanks.
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2. your diagram says 2.5 at the top and bottom but your post says 1.5?
3. oh thanks, edited it now.
4. Area of full page => hw=600
.:. h=600/w or w=600/h

Area of Printed=> A=(h-5)(w-3)
using h=600/w

A=(600/w -5)(w-3)
..=600-(1800/w) -5w +15
A=615-1800/w-5w

Double check that.
5. nice one Malik it seems to make good sense.
so what do i do for part (b)? differentiate? ^o)
6. yup, using second derivative to find max...
7. (Original post by Reloaded)
nice one Malik it seems to make good sense.
so what do i do for part (b)? differentiate? ^o)
b) dA/dw =615-1800w-1-5w
.............=1800/w²-5

max inner 1800/w²-5=0 => 1800/w²=5 => 5w²=1800 w²=360 w=6√10

h=600/w =600/6√10 =10√10

w=6√10
h=10√10

I am not sure if it wants that or the dimensions of the printed area
hp=10√10-5 wp=6√10-3
8. thanks guys, i'm sorry to say i dont really know how to do part (c) either. do i have to find the minimum aswell, as then draw a graph based on that?
9. A=615-1800/w-5w is what I get too.

therefore, dA/dw = 1800/W^2 - 5,

Let that equal 0.

1800 = 5W^2 and therefore, W = 6root10 and therefore (because Wh = 600), h = 100/root 10, 10 root 10. These are the dimensions for the page, overall.

The area (putting back the values I got above, into the equation) I get is 425. That is for the printed area.

The graph only has a maximum, because the second derivative is ,-3600/W^3, is always negative. Just find where to crosses the x-axis, ie. where 0 = 615 - 1800/W - 5W. And find out what the value of y is when x = 0. Using that, and the max point from the previous question, you can plot the graph you want.
10. the second derivative is ,-3600/W^3
i thought it was 1800/w^2-5 ?? - no wait thats F'(A), sorry

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Updated: December 11, 2005
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