Ask Question
15 May, 20:54

An open box is to be made out of a 6-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume

+1
Answers (1)
  1. 15 May, 21:06
    0
    The dimensions of the resulting box that has the largest volume is 1.3 inches x 1.3 inches

    Step-by-step explanation:

    Card board size is L = 14 inches and

    W = 6 inches

    Let x be the size of equal squares cut from 4 corners and bent into a box whose size is now;

    L = 14 - 2x, W = 6 - 2x and h = x inches.

    Volume of the box is given as;

    V = (14 - 2x) (6-2x) x

    V = (4x² - 40x + 84) x

    = 4x³ - 40x² + 84x.

    Now, for the maximum value,

    dV/dx = 0

    Thus,

    dv/dx = 12x² - 80x + 84 = 0

    Using quadratic formula

    x = [ - (-80) ± √ (-80²) - 4 (12 x 84) ] / (2 x 12)

    x = [80 ± √ (6400 - 4032) ]/24

    x = (80 + 48.66) / 24 or (80-48.66) / 24

    x = 5.36 or 1.31

    Looking at the two values, 1.31 would be more appropriate because if we use 5.36, we will get a negative value of the width (W).

    Thus, x = 1.31 inches

    Let us use the Second Derivative Test to verify that V has a local maximum at x = 1.31.

    Thus;

    d²v/dx² = 24x - 80 = 24 (1.31) - 80 = - 48.56

    This is less than 0 and therefore, the volume of the box is maximized when a 1.31 inch by 1.3 inch square is cut from the corners of the cardboard sheet.
Know the Answer?
Not Sure About the Answer?
Find an answer to your question 👍 “An open box is to be made out of a 6-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and ...” in 📗 Mathematics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers