Problem 57 - Precalculus: Mathematics for Calculus

Question

A specialty-car manufacturer has plants in Auburn, Biloxi, and Chattanooga. Three models are produced, with daily production given in the following matrix. $$\begin{array}{rccc} & \text { Cars produced each day } \\ \hline \text { Model K } & \text { Model R } & \text { Model W } \end{array}$$ $$\begin{aligned}&\begin{array}{c}\text { Auburn } \\ \text { Biloxi } \\\\\ \text { Chattanooga } \end{array}\left[\begin{array}{r} 12 & 10 & 0 \\ 4 & 4 & 20 \\ 8 & 9 & 12 \end{array}\right]=A\end{aligned}$$ Because of a wage increase, February profits are lower than January profits. The profit per car is tabulated by model in the following matrix. $$\begin{aligned}&\qquad\qquad\text { January } \quad \text { February }\\\&\begin{array}{c}\text { Model K }\\\ \text { Model R } \\ \text { Model W } \end{array}\left[\begin{array}{r}\(1000 & \)500 \\ \(2000 & \)1200 \\ \(1500 & \)1000 \end{array}\right]=B\end{aligned}$$ (a) Calculate \(A B\). (b) Assuming that all cars produced were sold, what was the daily profit in January from the Biloxi plant? (c) What was the total daily profit (from all three plants) in February?

Step-by-Step Solution

Answer

(a) Product \(AB = \begin{bmatrix} 32000 & 17000 \\ 36000 & 23700 \\ 49000 & 33400 \end{bmatrix}\); (b) 36,000; (c) 74,100.

Step 1: Understand the Problem

We need to compute the matrix product of the production matrix \(A\) with the profit matrices for January and February respectively, given as \(B\). This is a matrix multiplication problem where we combine the production quantities with profit value per model for each month.

Step 2: Matrix Multiplication Setup

The production matrix \(A\) and profit matrix \(B\) are given as: \[A = \begin{bmatrix} 12 & 10 & 0 \ 4 & 4 & 20 \ 8 & 9 & 12 \end{bmatrix}, B = \begin{bmatrix} 1000 & 500 \ 2000 & 1200 \ 1500 & 1000 \end{bmatrix}\]The resulting matrix \(AB\) will be a \(3 \times 2\) matrix. We will calculate each element of \(AB\).

Step 3: Calculate January Daily Profits

Use the given matrices to compute the elements of \(A B\) for January:For Auburn:\[ (12*1000 + 10*2000 + 0*1500, 12*500 + 10*1200 + 0*1000) = (32000, 17000) \]For Biloxi:\[ (4*1000 + 4*2000 + 20*1500, 4*500 + 4*1200 + 20*1000) = (36000, 23700) \]For Chattanooga:\[ (8*1000 + 9*2000 + 12*1500, 8*500 + 9*1200 + 12*1000) = (49000, 33400) \]So, for January, \[A B = \begin{bmatrix} 32000 & 17000 \ 36000 & 23700 \ 49000 & 33400 \end{bmatrix}\]

Step 4: Identify January Profit from Biloxi

The profit from Biloxi in January is the first element of the second row of the resultant matrix \(AB\) which we computed in the previous step. From \[A B\], it's 36,000.

Step 5: Calculate February Daily Profits Using the Same Method

We use the second column results (February) from our calculations in Step 3:Auburn: 17,000Biloxi: 23,700Chattanooga: 33,400Total February daily profit is:\( 17000 + 23700 + 33400 = 74100 \).

Step 6: Summarize the Calculated Values

We have calculated the profits as follows:(a) Product of matrices giving daily profits for January: \[ \begin{bmatrix} 32000 & 17000 \ 36000 & 23700 \ 49000 & 33400 \end{bmatrix} \](b) January profit from Biloxi: 36,000.(c) Total daily profit for February: 74,100.

Key Concepts

Production Matrix

A production matrix is a mathematical representation of the daily production at various manufacturing plants. Here, we focus on a specialty-car manufacturer with plants in Auburn, Biloxi, and Chattanooga, each producing different models of cars daily. The matrix encapsulates the number of each model manufactured per day at each plant, streamlining information:

• Auburn: 12 Model K, 10 Model R, and 0 Model W cars each day.
• Biloxi: 4 Model K, 4 Model R, and 20 Model W cars each day.
• Chattanooga: 8 Model K, 9 Model R, and 12 Model W cars each day.

The matrix simplifies complex production data, allowing easier calculations and comparisons. It's crucial in determining profits, as it contextualizes how many of each car model are produced, impacting overall revenue.

Profit Matrix

The profit matrix outlines the profit per car model for different months. It shows how changes in external factors like wage increases can affect profitability. In our example, we examine January and February's profit per car model:

• Model K: $1000 in January, $500 in February
• Model R: $2000 in January, $1200 in February
• Model W: $1500 in January, $1000 in February

The matrix is essential for understanding profit fluctuations. It gives direct insight into how model-specific profitability changes over time. Combining this matrix with the production matrix helps calculate actual profits during different months, highlighting the impact of economic changes on each car model.

Daily Profit Calculation

To determine daily profits, multiply the production matrix by the profit matrix. This gives a new matrix where each entry represents the total profit from all models produced at a plant for a specific month. Consider January's profits:

• Auburn: Calculated as \[(12 \times 1000 + 10 \times 2000 + 0 \times 1500, 12 \times 500 + 10 \times 1200 + 0 \times 1000) = (32000, 17000)\]
• Biloxi: \[(4 \times 1000 + 4 \times 2000 + 20 \times 1500, 4 \times 500 + 4 \times 1200 + 20 \times 1000) = (36000, 23700)\]
• Chattanooga: \[(8 \times 1000 + 9 \times 2000 + 12 \times 1500, 8 \times 500 + 9 \times 1200 + 12 \times 1000) = (49000, 33400)\]

For February's total daily profit, simply sum up the relevant column:\[17000 + 23700 + 33400 = 74100\]This process highlights the power of matrix multiplication in contextualizing complex data into actionable insights, streamlining calculations, and simplifying business analysis.