Answer:
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.
