2. Maximum Volume of a Box An Investigation

3. Maximum volume of a box From a square piece of cardboard of side 20 cm small corners of side x are cut off. 20cm x x x

4. The cardboard is then folded up to form a box as shown. Determine the value of x which will produce a box with the largest possible volume.

5. Solution Pupils will have to show that the volume of the box is given by V = 4x 3 - 80x 2 + 400x (ii) Press [Y= ] and enter the formula for the volume in Y 1 as shown

6. (iii) Press [2nd], [TBLSET] and set TblStart to 0 and ∆Tbl to 1 Pressing [2nd] and [TABLE] should produce the table

7. The maximum volume is to be found between x = 2 and x = 4 so we carry out a careful search between these values. (iv) Press [2nd], [TBLSET] and set TblStart to 2 and ∆Tbl to 0.2 Press [2nd], [TABLE] to display table 2 below

8. This table clearly shows that the maximum is to be found between x = 3.2 and x = 3.6. (v) Press [2nd], [TBLSET] and set TblSet to 3.2 and ∆Tbl to 0.1 Press [2nd], [TABLE] to produce the final table This part of the table shows that the maximum volume is 592.55 and is obtained when x = 3.3

9. Exercise:- Repeat the above calculations to find the box of maximum volume which can be made from a rectangular sheet of cardboard measuring 32cm by 22cm.