Question

If 12 ft2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

??

Answer 2

Let one side of the base be L.

Area of the base = L^2
Area of each side = (12-L^2)/4

Height = (area of side / width of side) = (12-L^2)/(4L)
Volume = B×h = (L^2) × (12-L^2)/(4L) = L×(12-L^2)/4
\[V = L(12-L^2)/4 \\ = -\frac{ 1 }{ 4 } L^3 + 3L\]
Now, what's the highest V can be (knowing that L is between 0 and sqrt 12)