We have a rectangular sheet of tinfoil whose dimensions are
32 centimetres by 20 centimetres.
Equal squares are cut
out at each of the corners.
Can you find the maximum volume of a wooden box which the
base and sides of the box?
Answer: 1152
cubic centimetres.
Let us assume v is the volume o f the box, and x is the side
of the squares cut out.
Then v = (32-2x) (20 – 2x) x
Or v= 640x – 104x²
+ 4x³
Dv / dx = 640 – 208x
+ 12x²
For a maximum volume
dv/dx = 0
Or 3x² - 52x + 160 = 0
Or (3x - 40) (x – 4) = 0
Or x= 4 or x= 40/3
The real volume of x can only be 4. Therefore the maximum
volume of the box is 24x 12 x a4 = 1152 cubic centimetres.
