Answer:
Max vol = 2 cubic metres
Step-by-step explanation:
Given that from a square piece of cardboard paper of area size 9 m2 , squares of the same size are cut off from each corner of the paper.
Side of the square = 3m
If squares are to be cut from the corners of the cardboard we have the dimensions of the box as
3-2x, 3-2x and x.
Hence x can never be greater than or equal to 1.5
V(x) = Volume = [tex]V(x) = x(3-2x)^2\\= 9x+4x^3-12x^2[/tex]
We use derivative test to find the maxima
[tex]V'(x) = 9+12x^2-24x\\V"(x) = 24x-24\\[/tex]
Equate I derivative to 0
[tex]9+12x^2-24x=0\\x=1/2,3/2[/tex]
If x= 3/2 box will have 0 volume
So this is ignored
V"(1/2) <0
So maximum when x =1/2
Maximum volume
=[tex]9(1/2)+4(1/2)^3-12(1/2)^2\\=2[/tex] cubic metres
