Answer:
2 feet
Step-by-step explanation:
We are given that
Length of piece,l=10ft
Width of piece,b=16ft
Let x be the side of square
Length of box=10-2x
Width of box=16-2x
Height of box=x
Volume of box=lbh
[tex]V=(10-2x)(16-2x)x[/tex]
[tex]V=x(160-52x+4x^2)=160x-52x^2+4x^3[/tex]
Differentiate w.r.t x
[tex]V'(x)=160-104x+12x^2[/tex]
[tex]V'(x)=0[/tex]
[tex]160-104x+12x^2=0[/tex]
[tex]3x^2-26x+40=0[/tex]
[tex]3x^2-20x-6x+40=0[/tex]
[tex]x(3x-20)-2(3x-20)=0[/tex]
[tex](3x-20)(x-2)=0[/tex]
[tex]x=\frac{20}{3},x=2[/tex]
[tex]x\neq \frac{20}{3}[/tex]
Because when substitute the value x=20/3
Then, the width of box is negative which is not possible.
[tex]V''(x)=-104+24 x[/tex]
Substitute x=2
[tex]V''(2)=-104+24(2)=-56<0[/tex]
Hence, the volume is maximum at x=2
Side of square=2 feet
Hence, square of side 2 feet should be cut from each corner in order to produce a box of maximum volume.
