Respuesta :
If you start with a 12x16 rectangle and cut square with side length x, when you bend the sides you'll have an inner rectangle with sides [tex]12-2x[/tex] and [tex]16-2x[/tex], and a height of x.
So, the volume will be given by the product of the dimensions, i.e.
[tex](12-2x)(16-2x)x = 4x^3-56x^2+192x[/tex]
The derivative of this function is
[tex]12x^2-112x+192[/tex]
and it equals zero if and only if
[tex]12x^2-112x+192=0 \iff x = \dfrac{14\pm 2\sqrt{13}}{3}[/tex]
If we evaluate the volume function at these points, we have
[tex]f\left(\dfrac{14-2\sqrt{13}}{3}\right) = \dfrac{64}{27}(35+13\sqrt{13})> f\left(\dfrac{14-2\sqrt{13}}{3}\right) = -\dfrac{64}{27}(13\sqrt{13}-35)[/tex]
So, the maximum volume is given if you cut a square with side length
[tex]x=\dfrac{14-2\sqrt{13}}{3}\approx 2.7[/tex]
Answer:
194.07 cubic inches.
Step-by-step explanation:
The cardboard is 12x16 before removing a square from each end. This square is x inches wide. Thus the 16 inside is shortened by x inches on both sides, or it is now 16-2x inches. The 12 inside is also reduced by 2x. The x value is also the height of the box when you fold the sides up. Thus the volume V = wlh = (16-2x)*(12-2x)*(x) = 4x^3 - 56x^2 + 192x.
To find the maximum, take the derivative, and find its roots
V = 4x^3 - 56x^2 + 192x
dV/dx = 12x^2 - 112x + 192
The roots are (14+2(13)^.5)/3 ~= 7.07 and (14-2(13)^.5)/3 ~= 2.26
The roots would be the possible values of x, the square we cut. Since 7.07 x 2 = 14.14 inches, this exceeds the 12 inch side, thus x = 2.26 inches. Thus you cut 2.26 inches from each corner to obtain the maximum volume.
Cube is 11.48 x 7.48 x 2.26 with a volume of 194.07 cubic inches.
