Problem 52 - College Algebra and Calculus: An Applied Approach

Question

A tire corporation has three factories, each of which manufactures two products. The number of units of product \(i\) produced at factory \(j\) in one day is represented by \(a_{i j}\) in the matrix \(A=\left[\begin{array}{rrr}80 & 120 & 140 \\ 40 & 100 & 80\end{array}\right]\) Find the production levels if production is decreased by 5\%. (Hint: Because a decrease of \(5 \%\) corresponds to \(100 \%-5 \%\), multiply the matrix by \(0.95 .\) )

Step-by-Step Solution

Answer

The new production levels after a 5% decrease are given by the following matrix: \( \left[\begin{array}{rrr} 76 & 114 & 133 \ 38 & 95 & 76 \end{array}\right]\)

Step 1: Understanding the problem

The problem involves applying a 5% decrease to the current production levels of each factory. In terms of matrix operations, this translates to multiplying the given matrix by the scalar 0.95.

Step 2: Perform the Scalar Multiplication

So, take the matrix of production levels \(A\) and multiply each entry by \(0.95\). This results in a new matrix that represents the new production levels. The computation is as follows: \(0.95 \times A= \left[\begin{array}{rrr} 0.95 \times 80 & 0.95 \times 120 & 0.95 \times 140 \ 0.95 \times 40 & 0.95 \times 100 & 0.95 \times 80 \end{array}\right]= \left[\begin{array}{rrr} 76 & 114 & 133 \ 38 & 95 & 76 \end{array}\right]\)

Step 3: Interpret the Result

Each entry in the resulting matrix provides the new production level for the respective product at each factory. Therefore, for example, after the 5% reduction in production levels, factory 1 would produce 76 units of product 1 and 38 units of product 2.

Key Concepts

Scalar Multiplication

Scalar multiplication is a basic operation in linear algebra involving a matrix and a number called a scalar. In this exercise, the matrix represents production levels across different factories and products, while the scalar is a factor that adjusts these levels by a percentage.

To perform scalar multiplication, multiply every element of the matrix by the scalar. In our case, we want to reduce production by 5%, which leaves 95% of the original production levels. We represent this as a scalar, 0.95.

For example, the matrix element for Factory 1, Product 1 is 80. Multiply 80 by 0.95 to get 76. Repeat this for each element in the matrix to find the new production levels. Scalar multiplication is a straightforward method to adjust all values in a matrix simultaneously, making it essential for many practical applications.

Production Levels

Production levels are a quantitative description of the output from factories or manufacturing units. In this problem, the production levels are detailed in a matrix with each element, denoted as \(a_{ij}\), representing the output of a specific product at a specific factory.

Understanding these levels is crucial for managing resources and planning. By organizing these values into a matrix, we can easily apply mathematical operations, like the scalar multiplication used to calculate decreased production.

Matrix representation of production levels helps in real-life scenarios by providing a clear overview at a glance. It enables easy adjustments through mathematical transformations, helping businesses make strategic decisions quickly and efficiently.

Matrix Interpretation

Matrix interpretation involves understanding the arrangement and meaning of numbers represented in a matrix. In our scenario, the matrix "A" provides the original daily production levels for two products across three factories.

After the scalar multiplication, each element in the matrix still represents specific production data: row 1 for Product 1 and row 2 for Product 2, across the columns which symbolize three factories.

For instance, the entry in the first row, first column shows that Factory 1 produces 76 units of Product 1 after a 5% decrease. Interpreting a production matrix correctly allows for informed decision-making in resource allocation and operational management. It serves as a bridge between raw numerical data and actionable insights.