Question

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

in., by cutting out square corners with side length x and folding up the sides.

Answer 1

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.

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.