Answer:
- width: 24 cm
- height: 36 cm
Explanation:
When margins are involved, the smallest area will be the one that has its dimensions in the same proportion as the margins. If x is the "multiplier", the dimensions of the printed area are ...
(4x)(6x) = 384 cm^2
x^2 = 16 cm^2 . . . . . divide by 24
x = 4 cm
The printed area is 4x by 6x, so is 16 cm by 24 cm. With the margins added, the smallest poster will be ...
24 cm by 36 cm
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Comment on margins
It should be obvious that if both side margins are 4 cm, then the width of the poster is 8 cm more than the printed width. Similarly, the 6 cm top and bottom margins make the height of the poster 12 cm more than the height of the printed area.
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Alternate solution
Let w represent the width of the printed area. Then the printed height is 384/w, and the total poster area is ...
A = (w+8)(384/w +12) = 384 +12w +3072/w +96
Differentiating with respect to w gives ...
A' = 12 -3072/w^2
Setting this to zero and solving for w gives ...
w = √(3072/12) = 16 . . . . matches above solution.
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Generic solution
If we let s and t represent the side and top margins, and we use "a" for the printed area, then the above equation becomes the symbolic equation ...
A = (w +s)(a/w +t)
A' = t - sa/w^2
For A' = 0, ...
w = √(sa/t)
and the height is ...
a/w = a/√(sa/t) = √(ta/s)
Then the ratio of width to height is ...
w/(a/w) = w^2/a = (sa/t)/a
width/height = s/t . . . . . . the premise we started with, above