Question

To manufacture an automobile requires painting, drying, and polishing. Epsilon Motor Company produces three types of cars: the Delta, the Beta, and the Sigma. Each Delta requires 10 hours (h) for painting, 3 h for drying, and \(2 \mathrm{~h}\) for polishing. A Beta requires \(16 \mathrm{~h}\) for painting, \(5 \mathrm{~h}\) for drying, and \(3 \mathrm{~h}\) for polishing, and a Sigma requires \(8 \mathrm{~h}\) for painting, \(2 \mathrm{~h}\) for drying, and \(1 \mathrm{~h}\) for polishing. If the company has \(240 \mathrm{~h}\) for painting, \(69 \mathrm{~h}\) for drying, and \(41 \mathrm{~h}\) for polishing per month, how many of each type of car are produced?

Short answer

7 Deltas, 3 Betas, and 8 Sigmas

Step-by-step solution

Define Variables

Let \(x\) be the number of Delta cars, \(y\) be the number of Beta cars, and \(z\) be the number of Sigma cars produced.

Create Equations for Painting Constraints

Each Delta requires 10 hours, each Beta requires 16 hours, and each Sigma requires 8 hours of painting. The total time available for painting is 240 hours. Therefore, the equation is: \[ 10x + 16y + 8z = 240 \]

Create Equations for Drying Constraints

Each Delta requires 3 hours, each Beta requires 5 hours, and each Sigma requires 2 hours of drying. The total time available for drying is 69 hours. Therefore, the equation is: \[ 3x + 5y + 2z = 69 \]

Create Equations for Polishing Constraints

Each Delta requires 2 hours, each Beta requires 3 hours, and each Sigma requires 1 hour of polishing. The total time available for polishing is 41 hours. Therefore, the equation is: \[ 2x + 3y + 1z = 41 \]

Solve the System of Equations

To find the number of each type of car produced, solve the system of linear equations: \[ \begin{cases} 10x + 16y + 8z = 240 \ 3x + 5y + 2z = 69 \ 2x + 3y + z = 41 \ \end{cases} \]

Use Methods to Solve the System

Use either substitution or Gaussian elimination method to solve the system. This step involves detailed calculations to find the values of \(x\), \(y\), and \(z\).

Verify the Solution

Ensure that the solution satisfies all three equations. This can be done by plugging the values of \(x\), \(y\), and \(z\) back into the original equations.

Key concepts

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear mappings between such spaces, and systems of linear equations.
It plays a crucial role in understanding and solving problems involving multiple variables and their linear relationships.
In the context of the problem given, understanding linear algebra allows us to frame the constraints provided by the resource limitations (painting, drying, polishing hours) as a system of linear equations.
Each equation represents a linear relationship between the different car types and the time taken for each task.
The system of linear equations that we then solve is:
\[ \begin{cases} 10x + 16y + 8z = 240 \ 3x + 5y + 2z = 69 \ 2x + 3y + z = 41 \ \end{cases} \]
This system represents the constraints in a mathematical way that can be solved using various methods.

Gaussian elimination

Gaussian elimination is a method for solving systems of linear equations.
It involves performing row operations to transform a matrix into a row-echelon form, from which the solutions to the equations can be easily obtained.
In this problem, Gaussian elimination can be used to solve the system of linear equations formed from the painting, drying, and polishing constraints.
The steps for Gaussian elimination include:

• Forming the augmented matrix for the system of equations.
• Applying row operations (swapping rows, multiplying rows by nonzero constants, adding or subtracting multiples of rows) to achieve an upper triangular form.
• Once in upper triangular form, back-substitution is used to solve for the variables starting from the bottom equation.

For our system, the augmented matrix would look like this initially: \[ \begin{pmatrix} 10 & 16 & 8 & \vert & 240 \ 3 & 5 & 2 & \vert & 69 \ 2 & 3 & 1 & \vert & 41 \ \end{pmatrix} \] By performing row operations, you systematically reduce it to solve for \( x, y, \text{and} z \).

substitution method

The substitution method is another way to solve systems of linear equations.
This method involves solving one of the equations for one variable in terms of the others and then substituting this expression into the remaining equations.
Here is how you can use the substitution method for this problem:

• First, solve one of the equations for one variable. For simplicity, let's solve the third equation for \(z\): \[ z = 41 - 2x - 3y \]
• Substitute \( z = 41 - 2x - 3y \) into the first and second equations.
• This substitution reduces the number of variables and thus the number of equations.
• Solve the resulting two-variable system for \(x\) and \(y\).
• Finally, use the values found for \(x\) and \(y\) to find \(z\).

Though substitution can be labor-intensive and prone to algebraic mistakes, it is straightforward and especially useful when the system is not too large.

constraint equations

Constraint equations are mathematical equations that represent the restrictions or limitations of a particular problem.
In our car manufacturing scenario, the constraint equations capture the limits on painting, drying, and polishing times.
The given constraints are:

• Painting: \( 10x + 16y + 8z = 240 \)
• Drying: \( 3x + 5y + 2z = 69 \)
• Polishing: \( 2x + 3y + z = 41 \)

These equations tell us that the combined workload for painting, drying, and polishing must equal the total available hours for each task.
This restriction forces us to find a balance in producing Delta, Beta, and Sigma cars.
Solving these constraints helps us determine the feasible number of each car type that can be manufactured without exceeding resource limits.