Answer :
Answer:
The maximum dimensions of the box:
Length of the box = 0.788676 m
Breadth of the box = 0.288676 m
Step-by-step explanation:
Original piece of cardboard is a square with sides of length s.
Length of the card board = l = 1 m
Breadth of cardboard = b = 1/2 m = 0.5 m
Squares with sides of length x are cut out of each corner of a rectangular cardboard to form a box.
Now, length of the box = L = 1 - 2x
And breadth of the box = B = 0.5 - 2x
Height of the box ,H = x
Volume of the box ,V= L × B × H
[tex]\frac{dV}{dx}=\frac{(1 -2x)(0.5-2x)x}{dx}[/tex]
[tex]\frac{dV}{dx}=\frac{d(0.5x-3x^2+4x^3)}{dx}[/tex]
[tex]\frac{dV}{dx}=0.5-6x+12x^2[/tex]
[tex]\frac{dV}{dx}=O[/tex]
x = 0.394338 , 0.105662
[tex]\frac{d^2V}{(dx)^2}=-6+24x[/tex]
When , [tex]x=0.105662[/tex] , [tex]\frac{d^2V}{(dx)^2}<0[/tex] (maxima)
The maximum dimensions of the box:
Length of the box = L = 1 - 2x = 1 - 2(0.105662) m = 0.788676 m
Breadth of the box = B = 0.5 - 2x = 0.5 - 2(0.105662) m = 0.288676 m