Determine the page dimensions that will minimize the amount of paper used?
A designer is designing a layout on a rectangular page that will contain 24 square inches of print. The margins at the top and bottom of the page will be 1.5 inches wide, and the margins on each side will be 1 inch wide.
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The area of the paper is of 24 square inches.
A = x*y
24 = x*y => y = 24/x
The dimensions of the rectangular layout are:
A(x) = (x+2)(y+3)
A(x) = (x+2)(24/x + 3)
To determine the dimensions that will minimize the amount of paper, we'll have to calcualte the 1st derivative of the area function.
We'll use the product rule:
A'(x) = (x+2)'*(24/x + 3) + (x+2)*(24/x + 3)'
A'(x) = 24/x + 3- 24(x+2)/x^2
A'(x) = (24x + 3x^2 - 24x - 48)/x^2
We'll eliminate like terms:
A'(x) = ( 3x^2 - 48)/x^2
We'll cancel A'(x);
A'(x) = 0
( 3x^2 - 48)/x^2 = 0 => 3x^2 - 48 = 0 => 3x^2 = 48 => x^2 = 16 => x1 = 4, x2 = -4
Since a dimension cannot be negative, we'll reject the negative value -4.
y = 24/4 => y = 6
Therefore, the minim dimensions of the paper are: x = 4 and y = 6.
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