An open box is made from a cardboard measuring 12cm x 12cm by cutting off equal squares from the corner and turning up the side. Then maximum possible volume of the box would be :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 128 cm3
Let the side of each square cut off be x cm. The base of the box is then (12 − 2x) cm square and its height is x cm, with 0 < x < 6.
\( V=x(12-2x)^2 \)
\( \displaystyle \frac{dV}{dx}=(12-2x)^2+x\cdot 2(12-2x)(-2) \)
\( \displaystyle =(12-2x)(12-2x-4x) \)
\( \displaystyle =(12-2x)(12-6x) \)
\( \displaystyle \frac{dV}{dx}=0\ \Rightarrow\ x=6 \text{ or } x=2 \)
x = 6 gives no box (V = 0), so take x = 2. For x slightly less than 2, \( \frac{dV}{dx} \) is positive, and for x slightly more than 2 it is negative, so V has a maximum at x = 2.
\( V=2(12-4)^2 \)
\( =2\times 64=128\ \text{cm}^3 \)
Shortcut: For a square sheet of side a, the volume is largest when the square cut off has side \( \frac{a}{6} \). Here x = 2, base 8 cm, volume \( 8^2\times 2=128 \) cm3.