A box without cover with a square base sides x and height h has a surface area of 12 m sq. Show that the volume of the box expressed in terms of x can be written as
V(x) = x(12-x^2)/4.
Determine the biggest volume such a box can have.

Question

A box without cover with a square base sides x and height h has a surface area of 12 m sq. Show that the volume of the box expressed in terms of x can be written as 
V(x) = x(12-x^2)/4. 
Determine the biggest volume such a box can have.

anonymous · Answer

surface area of base=x^2
surface area of sides=hx
4hx+x^2=12, 12-x^2=4hx, (12-x^2)/4x=h
volume of box=hx^2
V(x)=x^2((12-x^2)/4x), x(12-x^2)/4

anonymous · Answer

V(x)=3x-x^3/4
V'(x)=3-3x^2/4
V'(x)=0, 3x^2/4=3, x^2=4, x=2  (-2 is not applicable here)
max volume is when x=2, h=1

anonymous · Answer

Thanks a lot