Question

Make an open box from a 20cm by 20cm piece of cardboard by cutting out four squares
from corners folding the flaps. What is the biggest volume of box you can make in this way?
Can you find a relation between the size of paper and the size of the square cutout that
produces the maximum volume.
Extension: You can extend this by taking a rectangular sheet of paper instead of a square
sheet of paper.

Answer 1

Answer:

Let each side of the square cut off from each corner be x cm

Then the base of the box will be of side 18−2x cm and the height of the box will be x cm

Then volume of box V=(18−2x)(18−2x)x

V=(18−2x)

2

x

V=4x

3

+324x−72x

2

...(i)

Differentiating w.r t to x, we get

dx

dV

=12x

2

+324−144x

dx

dV

=12(x

2

−12x+27) ....(ii)

For maximum volume

dx

dV

=0

⇒12(x

2

−12x+27)=0

⇒ x

2

−9x−3x+27=0

⇒ (x−9)(x−3)=0

⇒ x=9,3

Again differentiating, we get

dx

2

d

2

V

=2x−12 ...(iii)

At x=9,

dx

2

d

2

V

=+ve

∴ V is minimum at x=9 at x=3

dx

2

d

2

V

=−ve

∴ V is maximum at x=3

∴ Maximum volume V=(18−6)(18−6)×3

=12×12×3=432cm

3

Answer 2

hope you got the answer