Question

A square piece of tin of side 18cm is to be made into a box without top by cutting a square from each corner and folding up the flops to form the box. The side of the square so that

the volume of the box is the maximum possible is given by

• A. 9
• B. 6
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

Volume $(18 - 2x)^{2}.x$

$4(9 - x)^{2}.x$

= $4\left( 81 + x^{2} - 18x \right)x$

$V = 4\left( x^{3} - 18x^{2} - 81x \right)$

⇒ $\frac{dV}{dx} = 4\left( 3x^{2} - 36x + 81 \right) = 0$

$x^{2} - 12x + 27 = 0$ ⇒ $x = 9,3$

$\frac{d^{2}V}{dx^{2}} = 4(6x - 36)$ ⇒ $\left. \ \frac{d^{2}V}{dx^{2}} \right|_{x = 3} - ve$

⇒ x = 3 so 'C' is correct.