Question

A rectangular sheet of metal has identical squares cut from each corner. The sheet is then bent along the dotted lines to form an open box. The volume of the box is 420 in.3.

The equation 4x3 – 72x2 + 320x = 420 can be used to find x, the side length of the square cut from each corner.

What is the side length of the square that is cut from each corner, to the nearest inch?

Answer 1

The length of the side is 3 inches.

Explanation:

Given : The volume of the box is 420 inches cube. The equation
`4x^3-72x^2 + 320x = 420` can be used to find x, the side length of the square cut from each corner.

To find : The side length of the square that is cut from each corner, to the nearest inch.

Solution :

To solve the equation we solve it using graph.

We plot the graph of the equation
`y=4x^3-72x^2 + 320x` and y=420

The intersection of these points would be the side of the length.

The intersection points are (2.89,420) ,(3,420) and (12.11,420)

We see that the roots of the equation is (0,0) , (8,0), (10,0)

There the factored form is

`y=4x^3-72x^2 + 320x=x(10-x)(8-x)`

`4x^3-72x^2 + 320x = 420`

`4x(10-x)(8-x)=420`

`x(10-x)(8-x)=105`

`x(10-x)(8-x)=3(7)(5)`

Thus comparing we get x=3

Therefore, The length of the side is 3 inches.

Answer 2

The correct answer is:

3 inches.

Explanation:

We will use the Rational Roots Theorem to solve this.

First we can divide both sides of the equation by 4 in order to simplify it:

`image`

In order to solve this, we want the polynomial set equal to 0. To do this, subtract 105 from both sides:

`image`

The Rational Roots Theorem says that if p/q is a root of the polynomial, then p is a factor of the constant term and q is a factor of the leading coefficient. The constant term is -105. Drawing a factor tree, we find that the factors of this number are 1, -1, 3, -3, 5, -5, 7, -5, 15, -15, 21, -21. The leading coefficient is 1; its only factor is 1. This means p/q must be a whole number, and can be any of the factors of 105.

Using synthetic division, we try 1 in the box:

1 | 1 -18 80 -105

_______ 1__ -17___63__

1 -17 63 -42

Since there is a remainder, this is not a root. Trying -1,

-1 | 1 -18 80 -105

_______-1__ 19 _-99_

1 -19 99 -204

This is not a root; in fact, it shows us that none of the negatives will be a factor, as the absolute values increase as we complete the synthetic division.

Trying 3,

3 | 1 -18 80 -105

________3__-45___105_

1 -15 35 0

Since there is no remainder, 3 is a root, and is the answer we are looking for.