A box open from top is made from a rectangular sheet of dimension a [] b by cutting square…

Question

A box open from top is made from a rectangular sheet of dimension a $$\times$$ b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

Accepted Answer

V = l . b . h = (a - 2x)(b - 2x) x

$$\Rightarrow$$ V(x) = (2x - a)(2x - b) x

$$\Rightarrow$$ V(x) = 4x3 - 2(a + b)x2 + abx

$$ \Rightarrow {d \over {dx}}v(x) = 12{x^2} - 4(a + b)x + ab$$

$${d \over {dx}}(v(x)) = 0 \Rightarrow 12{x^2} - 4(a + b)x + ab = 0 < _\beta ^\alpha $$

$$ \Rightarrow x = {{4(a + b) \pm \sqrt {16{{(a + b)}^2} - 48ab} } \over {2(12)}}$$

$$ = {{(a + b) \pm \sqrt {{a^2} + {b^2} - ab} } \over 6}$$

Let $$x = \alpha  = {{(a + b) + \sqrt {{a^2} + {b^2} - ab} } \over 6}$$

$$\beta  = {{(a + b) - \sqrt {{a^2} + {b^2} - ab} } \over 6}$$

Now, $$12(x - \alpha )(x - \beta ) = 0$$

$$\therefore$$ x = $$\beta$$ $$ = {{a + b - \sqrt {{a^2} + {b^2} - ab} } \over b}$$

Solutions · 1