Answer:
length of box = 13 - 2 (1.61) = 9.78 ft
width = 8 - 2(1.61) = 4.78 ft
height, x = .61 ft
Step-by-step explanation:
Length of sheet, L = 13 ft
width of sheet, W = 8 ft
height of the box, h = x
So, after cutting the sides, the length of the box is (13 -2x) and the width is (8 - 2x) and height is x.
Volume of the box = length x width x height
V = (13 - 2x)(8 - 2x) x
V = 4x³ - 42x² + 104 x
Differentiate with respect to x.
dV/dx = 12x² - 84 x + 104
Put it equal to zero for maxima and minima:
12x² - 84 x + 104 = 0
3x² - 21 x + 26 = 0
[tex]x=\frac{21\pm \sqrt{441-312}}{6}[/tex]
[tex]x=\frac{21\pm 11.36}{6}[/tex]
x = 5.4 ft or 1.61 ft
[tex]\frac{d^{2}V}{dx^{2}}=24x - 84[/tex]
for x = 5.4, d²V/dx² = 24(5.4) - 84 = + 45.6
for x = 1.61 , d²V/dx² = 24(1.61) - 84 = - 45.4
So, for x = 1.61 the volume is maximum.
length of box = 13 - 2 (1.61) = 9.78 ft
width = 8 - 2(1.61) = 4.78 ft
height, x = .61 ft
