Question

1. A box without a top is to be made from a rectangular piece of cardboard, with dimensions 3 in. b)

Answer 1

Answer:

a) The volume of the box is represented by $V = 21\cdot x - 20\cdot x^{2}+4\cdot x^{3}$.

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 ($V$), in cubic inches, is modelled by the equation for the cuboid:

$V = (3-2\cdot x) \cdot (7-2\cdot x) \cdot x$ (1)

Where $x$ is the side length of the cutted square corners, in inches.

$V = (21 - 20\cdot x + 4\cdot x^{2})\cdot x$

$V = 21\cdot x - 20\cdot x^{2}+4\cdot x^{3}$

The volume of the box is represented by $V = 21\cdot x - 20\cdot x^{2}+4\cdot x^{3}$.

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.