Answer: V(W) = (1/3)*(*W^2*800ft - 8W^3) and the domain is 0 < W < 100ft.
Step-by-step explanation:
The dimensions of the box are:
L = length
W = width
H = heigth.
We know that:
L = 4*W
And the girth of a box is equal to: G = 2*W + 2*H
then we have:
2*W + 2*H + H = 200ft
2W + 3*H = 200ft
Then we have two equations:
L = 4*W
2W + 3*H = 200ft
We want to find the volume of the box, which is V = W*L*H
and we want in on terms of W.
Then, first we can replace L by 4*W (for the first equation)
and:
2*W + 3*H = 200ft
3*H = 200ft - 2*W
H = (200ft - 2*W)/3.
then the volume is:
V(W) = W*(4*W)*(200ft - 2*W)/3
V(W) = (1/3)*(*W^2*800ft - 8W^3)
The domain of this is the set of W such that the volume is positive, then we must have that:
W^2*800ft > 8W^3
To find the maximum W we can see the equality (the minimum extreme is 0 < W, because the width can only be a positive number)
W^2*800ft = 8W^3
800ft = 8*W
100ft = W.
This means that if W is equal or larger than 100ft, the equation gives a negative volume.
Then the domain is 0 < W < 100ft.
