Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Inventory Levels A company sells five different models of computers through three retail outlets. The inventories of the five models at the three outlets are given by the matrix \(S\). $$ S=\left[\begin{array}{lllll} 3 & 2 & 2 & 3 & 0 \\ 0 & 2 & 3 & 4 & 3 \\ 4 & 2 & 1 & 3 & 2 \end{array}\right] $$ The wholesale and retail prices for each model are given by the matrix \(T\). $$ T=\left[\begin{array}{rl} \$ 900 & \$ 1200 \\ \$ 1200 & \$ 1450 \\ \$ 1400 & \$ 1650 \\ \$ 2650 & \$ 3250 \\ \$ 3050 & \$ 3375 \end{array}\right] $$ (a) What is the total retail price of the inventory at Outlet \(1 ?\) (b) What is the total wholesale price of the inventory at Outlet 3 ? (c) Compute the product \(S T\) and interpret the result in the context of the problem.

Step-by-Step Solution

Answer

The total retail price of inventory at Outlet 1 calculated from the matrix multiplication and the total wholesale price of the inventory at Outlet 3 will be found in the same way. The resulting matrix from the multiplication \(S * T\) gives the total prices for each computer model in each retail outlet.

Step 1: Calculate the total retail price at Outlet 1

To calculate the total retail price at Outlet 1, you should multiply the first row of matrix S with the second column of matrix T. This is done by multiplying each element in the row of matrix S with the corresponding element in the column of matrix T and then summing up these products.

Step 2: Calculate the total wholesale price at Outlet 3

Similar to Step 1, but now the calculation is for the wholesale price at Outlet 3. So, the third row of matrix S should be multiplied with the first column of matrix T.

Step 3: Compute the product S * T

Multiplying two matrices means to multiply each row in the first matrix (matrix S in this case) with each column in the second matrix (T). The product of the matrices S and T will be another matrix.

Step 4: Interpretation of the resulting Matrix

The resulting matrix from the multiplication of S and T will represent the total prices at the outlets; each element in the matrix represents a total price value for a certain computer model in a certain outlet.

Step 5: Calculation of respective total prices

By performing the calculation in the steps one and two, as well as considering the number of each computer model in each outlet, you get the sums of retail and wholesale prices for the computer models at the respective outlets.

Key Concepts

Retail Price Calculation

Calculating the retail price involves understanding how each computer model's price contributes to the total amount at a retail outlet. Imagine a store where several models of computers are stocked. Each model has a specific retail price which is the amount customers pay to purchase these computers.

To determine the retail price across all models at a particular outlet, we can use matrix multiplication. This involves multiplying the inventory levels (matrix S) by the retail prices (matrix T).

• Identify the row that corresponds to the outlet's inventory for each model.
• Multiply each inventory count by its corresponding retail price.
• Add up these values to find the total retail price for the outlet.

This process provides a clear financial overview by showing the total potential income from retail sales at a given outlet, and gives insight into pricing strategies and inventory prioritization.

Wholesale Price Calculation

Wholesale prices are what the retailer pays the supplier or manufacturer for each model of computer. It's crucial to calculate these when managing costs effectively. Like the retail price calculation, we use matrix multiplication to find the wholesale price at an outlet.

For Outlet 3, the approach is these simple steps:

• Identify the row representing Outlet 3's inventory levels in matrix S.
• Multiply these inventory levels with the wholesale prices in matrix T's first column.
• Sum the resulting values to get the total wholesale cost for Outlet 3.

Knowing this information helps businesses manage their initial purchasing costs, and makes it easier to see which outlet requires stock replenishment or how to allocate budget for new inventory.

Inventory Management

Inventory management is the process of ensuring that a business maintains the optimal number of products. It involves balancing too much inventory (leading to high storage costs) with too little (leading to potential stockouts and lost sales).

Using matrices like in this exercise provides a structured way to encapsulate inventory data cross-referenced with financial details like cost and price.

• It enables quick calculations of total prices across different models and outlets.
• Matrices reveal patterns in inventory distribution across stores.
• They allow for scenario planning and financial forecasting.

This approach essentially aligns inventory levels with business objectives, ensuring that each outlet has enough stock of each model to meet customer demand while managing costs efficiently.