Problem 65 - Precalculus with Limits

Question

MANUFACTURING A corporation has three factories,each of which manufactures acoustic guitars and electric guitars. The number of units of guitars produced at factory \(j\) in one day is represented by \(a_{ij}\) in the matrix \(A=\left[\begin{array}{r} 70 & 50 & 25 \\ 35 & 100 & 70 \end{array}\right]\) Find the production levels if production is increased by \(20\%\).

Step-by-Step Solution

Answer

The production levels increased by 20% are represented by the matrix \(B = \left[\begin{array}{ccc} 84 & 60 & 30 \ 42 & 120 & 84 \end{array}\right]\).

Step 1: Understand the Matrix

The given matrix 'A' represents the production levels of two types of guitars in three factories. Each element in the matrix \(a_{ij}\) represents the number of units produced in a day. The first row corresponds to the production of acoustic guitars and the second row to electric guitars. The columns represent the three factories. Thus, for instance, \(a_{11}\) corresponds to the number of acoustic guitars produced in factory 1, and \(a_{32}\) corresponds to the number of electric guitars produced in factory 2.

Step 2: Calculate the Increase in Production

The problem asks for the production levels if production is increased by 20%. To calculate this, you should multiply each element in the matrix by 1.2. The multiplication of a matrix by a scalar (in this case 1.2) is done element-wise. In other words, every element of the matrix is multiplied by the scalar. So the new matrix will be \(B = 1.2A\), where B will be the new matrix with the increased production levels.

Step 3: Perform the Matrix Multiplication

Perform the multiplication \(B = 1.2A\) to get the new matrix B which represents the increased production levels. This calculation will be performed as follows: \(B = 1.2 * \left[\begin{array}{ccc} 70 & 50 & 25 \ 35 & 100 & 70 \end{array}\right] = \left[\begin{array}{ccc} 1.2*70 & 1.2*50 & 1.2*25 \ 1.2*35 & 1.2*100 & 1.2*70 \end{array}\right]\).

Step 4: Calculate the New Matrix

Now perform the multiplication for each individual element and formulate the new matrix: \(B = \left[\begin{array}{ccc} 84 & 60 & 30 \ 42 & 120 & 84 \end{array}\right]\). So this is the matrix representing the increase in production by 20% for each guitar in each factory.

Key Concepts

Scalar Multiplication

Scalar multiplication is a straightforward mathematical operation where each element of a matrix is multiplied by a single number, known as a scalar. Let's delve into why this is important and how it relates to our exercise.
Imagine you have a matrix that represents certain quantities, such as production numbers. You want to increase these values by a consistent percentage, say 20%. This is where scalar multiplication comes in handy.

• In scalar multiplication, each element of the matrix is multiplied by this scalar, which, in our case, is 1.2 (because a 20% increase translates to multiplying by 1 + 0.2 = 1.2).
• The operation is performed individually on each element of the matrix, ensuring that the relative magnitude and proportions remain consistent within the matrix itself. This is crucial in operations like scaling production across factories.

By applying this concept to our exercise, we ensure that each factory's production capability is boosted evenly, maintaining balance across the board.

Matrices

Matrices are a fundamental concept in mathematics, acting like convenient containers for storing data that can represent anything from scientific measurements to production levels at a factory.
In a matrix, data is organized in a rectangular format comprising rows and columns, making it easy to manage and perform calculations such as scalar multiplication.
For instance, in our example, the given matrix:

• Rows represent different types of guitars (acoustic and electric).
• Columns represent the specific factories where these guitars are produced.

This arrangement allows us to keep track of how many guitars each type is producing per factory. It simplifies understanding different production levels, ensuring that each factory's output can be easily analyzed, adjusted, and optimized. The structure of a matrix enhances the ability to perform many types of mathematical operations, especially those involving complex data sets.

Production Levels

Understanding production levels is crucial when analyzing manufacturing processes and forecasting production capabilities. matrices provide a concise way to express these levels across various locations or types of products.
Consider how the matrix in this exercise enables us to see at a glance:

• The daily output of both acoustic and electric guitars.
• Performance comparisons across the three different factories.

When the production increase is applied through scalar multiplication, we get a new matrix detailing further enhanced production levels. By multiplying the original matrix by 1.2, we can easily project the new output levels for each type of guitar at each factory. This data-driven approach aids in decision-making and strategic planning, ensuring production efficiency and addressing any potential shortfalls or surpluses. With these insights, manufacturers can allocate resources more effectively and streamline operations for improved outcomes.