Calculating Widgets Produced Per Hour

Oct 13, 2025 by miodeya.com 18 views

Introduction

In this article, we will explore the concept of calculating the number of widgets produced per hour by each of three widget-producing machines. We will use a table to represent the running time, in hours, for each machine over the course of three months. The table will provide us with the necessary information to calculate the number of widgets produced per hour by each machine.

Table: Running Time for Each Machine

Machine	Month 1	Month 2	Month 3	Total Hours
1	120	130	140	390
2	110	120	130	360
3	100	110	120	330

Calculating Widgets Produced Per Hour

To calculate the number of widgets produced per hour by each machine, we need to first calculate the total number of widgets produced by each machine over the course of three months. Let $x_1, x_2, \text{ and } x_3$ represent the number of widgets produced by machines 1, 2, and 3, respectively.

We can represent the total number of widgets produced by each machine as follows:

$x_1 = 120 \times w_1 + 130 \times w_2 + 140 \times w_3$ $x_2 = 110 \times w_2 + 120 \times w_2 + 130 \times w_3$ $x_3 = 100 \times w_3 + 110 \times w_2 + 120 \times w_3$

where $w_1, w_2, \text{ and } w_3$ represent the number of widgets produced per hour by machines 1, 2, and 3, respectively.

Solving for $w_1, w_2, \text{ and } w_3$

To solve for $w_1, w_2, \text{ and } w_3$ , we can use the following system of equations:

$\begin{bmatrix} 120 & 130 & 140 \\ 110 & 120 & 130 \\ 100 & 110 & 120 \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$

We can solve this system of equations using various methods, such as substitution or elimination.

Using Substitution Method

Let's use the substitution method to solve for $w_1, w_2, \text{ and } w_3$ . We can start by solving the first equation for $w_1$ :

$120w_1 + 130w_2 + 140w_3 = x_1$

$w_1 = \frac{x_1 - 130w_2 - 140w_3}{120}$

We can substitute this expression for $w_1$ into the second equation:

$110w_2 + 120w_2 + 130w_3 = x_2$

$110w_2 + 120(\frac{x_1 - 130w_2 - 140w_3}{120}) + 130w_3 = x_2$

Simplifying this equation, we get:

$110w_2 + \frac{x_1 - 130w_2 - 140w_3}{10} + 130w_3 = x_2$

We can solve this equation for $w_2$ :

$w_2 = \frac{x_2 - \frac{x_1 - 140w_3}{10} - 130w_3}{110}$

We can substitute this expression for $w_2$ into the third equation:

$100w_3 + 110w_2 + 120w_3 = x_3$

$100w_3 + 110(\frac{x_2 - \frac{x_1 - 140w_3}{10} - 130w_3}{110}) + 120w_3 = x_3$

Simplifying this equation, we get:

$100w_3 + \frac{x_2 - \frac{x_1 - 140w_3}{10} - 130w_3}{10} + 120w_3 = x_3$

We can solve this equation for $w_3$ :

$w_3 = \frac{x_3 - \frac{x_2 - \frac{x_1 - 140w_3}{10} - 130w_3}{10} - 120w_3}{100}$

Solving for $w_1, w_2, \text{ and } w_3$

We can substitute the expressions for $w_2$ and $w_3$ into the expression for $w_1$ :

$w_1 = \frac{x_1 - 130(\frac{x_2 - \frac{x_1 - 140w_3}{10} - 130w_3}{110}) - 140(\frac{x_3 - \frac{x_2 - \frac{x_1 - 140w_3}{10} - 130w_3}{10} - 120w_3}{100})}{120}$

Simplifying this expression, we get:

$w_1 = \frac{x_1 - \frac{130x_2 - 130\frac{x_1 - 140w_3}{10} - 130^2w_3}{110} - \frac{140x_3 - 140\frac{x_2 - \frac{x_1 - 140w_3}{10} - 130w_3}{10} - 140 \cdot 120w_3}{100}}{120}$