Respuesta :
Answer:
Height = 3 cm, Length = 9 cm, Width = 9 cm
Step-by-step explanation:
A square piece of sheet of side x centimeters is cut from the square sheet. When it was folded to make the box the height of box becomes x, length becomes (18-2x) and the width becomes (18-2x).
Volume is given by
[tex]V = Length\times Width\times Height\\V = (18 - 2x)^2x = x(4x^2-72x+324) = 4x^3-72x^2+324x\\So,\\V(x) = 4x^3-72x^2+324x[/tex]
First, we differentiate V(x) with respect to x, to get,
[tex]\dfrac{d(V(x))}{dx} = \dfrac{d(4x^3-72x^2+324x)}{dx} = 12x^2 - 144x +324[/tex]
Equating the first derivative to zero, we get,
[tex]\dfrac{d(V(x))}{dx} = 0\\\\12x^2 - 144x +324 = 0[/tex]
Solving, with the help of quadratic formula, we get,
[tex]x =9,x=3[/tex]
Again differentiation V(x), with resopect to x, we get,
[tex]\dfrac{d^2(V(x))}{dx^2} = 24x - 144[/tex]
At x = 9
[tex]\dfrac{d^2(V(x))}{dx^2} > 0[/tex]
At x = 3
[tex]\dfrac{d^2(V(x))}{dx^2} < 0[/tex]
Thus, by double differentiation test, the maxima occurs at x = 3 for V(x).
Thus, largest volume the box can have occurs when x = 3.
Dimensions of box:
Height = x = 3 cm
Length = (18-2x) = 9 cm
Width = (18-2x) = 9 cm
