Question 1057989
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The top and bottom margins of a poster are 6 cm and the side margins are each 6 cm. If the area of printed material on the poster 
is fixed at 384 square centimeters, find the dimensions of the poster with the smallest area.
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Let x be the length of the printed material, in centimeters.
Then the wide is {{{384/x}}} centimeters.

Accounting for the margins, the area of poster is

S = (x + 2*6)*(384/x + 2*6),  or

S = (x+12)*(384/x + 12),  or

S = {{{384 + 12*(384/x) + 12x + 144}}}.   (3)

The problem asks to minimize S as a function of "x".

To do it, take the derivative {{{(dS)/(dx)}}} of (3).

{{{(dS)/(dx)}}} = -12*(384/x^2) + 12.

The condition {{{(dS)/(dx)}}} = 0 leads to the equation

{{{384/x^2}}} = 1,   or

{{{x^2}}} = {{{384}}}.

Then x = {{{sqrt(384)}}} = 19.6 cm is the solution (approximately).


<U>Answer</U>.  The poster's dimensions providing minimum area with the given text area and given margins 

         are 19.6+12 = 31.6 and {{{384/19.6+12}}} centimeters.
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