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26 March, 06:47

A rectangular page is to contain 24 sq. in. of print. The margins at the top and bottom of the page are each 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the page be so that the least amount of paper is used

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  1. 26 March, 09:42
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    Dimensions of page should be width of 6 inches and height of 9 inches

    Step-by-step explanation:

    Let x be the width of the printed part in inches

    Let y be height of the printed part in inches.

    Thus, Area of printed part; A = xy

    And area of printed part is given as 24.

    Thus, xy = 24

    Making y the subject, we have;

    y = 24/x

    Now, the question says the top and bottom margins are 1.5 inches.

    Thus, width of page = x + 1 + 1 = x + 2

    And also the margins on each side are both 1m in length, thus the height of page will be:

    y + 1.5 + 1.5 = y + 3

    So area of page will now be;

    A = (x + 2) • (y+3)

    From earlier, we got y = 24/x

    Thus, plugging this into area of page, we have;

    A = (x + 2) • ((24/x) + 3)

    A = 24 + 3x + 48/x + 6

    A = 30 + 3x + 48/x

    For us to find the minimum dimensions, we have to find the derivative of A and equate to zero

    Thus,

    dA/dx = 3 - 48/x²

    Thus, dA/dx = 0 will be

    3 - 48/x² = 0

    Multiply through by x²:

    3x² - 48 = 0

    Thus,

    3x² = 48

    x² = 48/3

    x = √16

    x = 4 inches

    Plugging this into y = 24/x, we have;

    y = 24/4 = 6 inches

    We want dimensions of page at x = 4 and y = 6.

    From earlier, width of page = x + 2.

    Thus, width = 4 + 2 = 6 inches

    Height = y + 3 = 6 + 3 = 9 inches

    So dimensions of page should be width of 6 inches and height of 9 inches
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