Answer:
a) The volume of the box is represented by [tex]V = 21\cdot x - 20\cdot x^{2}+4\cdot x^{3}[/tex].
b) A side length of 0.653 inches leads to the maximum volume of the box: 6.299 inches.
Step-by-step explanation:
a) The volume of the box ([tex]V[/tex]), in cubic inches, is modelled by the equation for the cuboid:
[tex]V = (3-2\cdot x) \cdot (7-2\cdot x) \cdot x[/tex] (1)
Where [tex]x[/tex] is the side length of the cutted square corners, in inches.
[tex]V = (21 - 20\cdot x + 4\cdot x^{2})\cdot x[/tex]
[tex]V = 21\cdot x - 20\cdot x^{2}+4\cdot x^{3}[/tex]
The volume of the box is represented by [tex]V = 21\cdot x - 20\cdot x^{2}+4\cdot x^{3}[/tex].
b) The method consist in graphing the polynomial and looking for a relative maximum. We graph the equation found in a) by means of a graphic tool. We present the outcome in the image attached below. According to this, a side length of 0.653 inches leads to the maximum volume of the box: 6.299 inches.
