Activity 8: Maximum Volume of an Open Box

Posted on May 7, 2024 in Physics

Objective:

To construct an open box with the maximum volume from a given rectangular sheet by cutting equal squares from each corner.

Method of Construction:

  1. Take a rectangular chart paper of size 20cm x 10cm and name it as ABCD.
  2. Cut four equal squares, each of side x cm, from each corner A, B, C, D.
  3. Repeat the process with the same size chart papers and different values of x.
  4. Make an open box by folding its flaps using sellotape.

Demonstration:

  1. When x=1, Volume=144 cm3
  2. When x=1.5, Volume=178.5 cm3
  3. When x=1.8, Volume=188.9 cm3
  4. When x=2, Volume=192 cm3
  5. When x=2.1, Volume=192.4 cm3
  6. When x=2.2, Volume=192.2 cm3
  7. When x=2.5, Volume=187.5 cm3
  8. When x=3, Volume=168 cm3

Clearly, the volume of the box is maximum when x=2.1 cm.

Observation:

  1. V1 = Volume (when x=1.6)
  2. V2 = Volume (when x=1.9)
  3. V3 = Volume (when x=2.1)
  4. V4 = Volume (when x=2.2)
  5. V5 = Volume (when x=3.2)
  6. Volume V1 is less than volume V3.
  7. Volume V2 is less than volume V3.
  8. Volume V3 is greater than volume V1, V2, V4, and V5.
  9. Volume V4 is less than volume V3.
  10. Volume V5 is less than volume V3.

Therefore, the volume of the open box is maximum when x = 2.1 cm.

Application:

This activity is useful in explaining the concepts of maxima and minima of functions. It is also useful in making packages with maximum volume at minimum cost.