A box open from top is made from a rectangular sheet of dimension a × 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
(1) \(\frac{a+b - \sqrt{a^2 + b^2 - ab}}{12}\)
(2) \(\frac{a+b - \sqrt{a^2 + b^2 + ab}}{6}\)
(3) \(\frac{a+b - \sqrt{a^2 + b^2 - ab}}{6}\)
(4) \(\frac{a+b - \sqrt{a^2 + b^2 - ab}}{6}\)
Correct answer: (3) \(\frac{a+b - \sqrt{a^2 + b^2 - ab}}{6}\)
Explanation:
V = ℓ. b. h = ( a – 2x) ( b – 2x) x
⇒ V(x) = (2x – a) ( 2x – b) x
⇒ V(x) = 4x3 – 2 ( a + b) x2 + abx
⇒ \(\frac{d}{dx}\)v(x) = 12x2 – 4 (a + b) x + ab
\(\frac{d}{dx}\)v(x) = 0 ⇒ 12x2 – 4 (a + b) x + ab = \(0<_{\beta}^{\alpha}\)
⇒ \(\frac{4(a+b) \pm \sqrt{16(a+b)^2 - 48ab}}{2(12)}\)
= \(\frac{(a+b) \pm \sqrt{a^2 + b^2 - ab}}{6}\)
Let x = α = \(\frac{(a+b) + \sqrt{a^2 + b^2 - ab}}{6}\)
β = \(\frac{(a+b) - \sqrt{a^2 + b^2 - ab}}{6}\)
Now, 12(x - α) (x - β) = 0
∴ x = β
= \(\frac{(a+b) - \sqrt{a^2 + b^2 - ab}}{b}\)