CAT 2003 Slot 2 — QA Question 5
A square tin sheet of side 12 inches is converted into a box with open top in the following steps – the sheet is placed horizontally. Then, equal sized squares, each of side x inches, are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of a box. If x is an integer, then what value of x maximizes the volume of the box?
Answer & solution
- A
3
- B
4
- C
1
2
When the tin sheet is cut across its corners as shown in the figure, the box formed will have a height of x inches and its base will be a square of side (12 – 2x) inches.
Let the volume of the box, V = (12 – 2x)2 × x = 4x3 − 48x2 + 144x
For V to be maximum, shoule be 0.
i.e. 12x2 − 96x + 144 = 0
∴ 12(x − 6)(x − 2) = 0
∴ x = 2 or x = 6
However, x cannot be 6 as the length of the side is (12 – 2x).
∴ x = 2
Hence, option (d).
Alternatively,
Since V = (12 – 2x)2 × x = [2(6 – x)]2 × x = 4x(6 – x)2,
Substituting values of x from 1 to 5, we get V maximum when x = 2 (i.e. V = 128)