Question

A vacation service has identified four resort hotels with a special all- inclusive package (room and meals included) at a popular travel destination. The quoted room rates are for double and family (maximum of four people) occupancy for 5 days and 4 nights. The current rates for the two types of rooms at the four hotels are represented by the matrix \(A\). $$A=\left[\begin{array}{rrrr}615 & 670 & 740 & 990 \\ 995 & 1030 & 1180 & 1105\end{array}\right]$$ If room rates are guaranteed not to increase by more than \(12 \%\) by next season, what is the maximum rate per package per hotel?

Short answer

The maximum rate per package per hotel for the next season, considering a 12% increase, would be determined by the maximum rate obtained from the matrix \(B\).

Step-by-step solution

Identify the matrix

The two types of rooms at the four hotels are represented by the following matrix: \(A=\left[\begin{array}{rrrr}615 & 670 & 740 & 990 \ 995 & 1030 & 1180 & 1105\end{array}\right]\).

Calculate the increased room rates

The maximum increase in room rates is 12 %. To get the increased rates, multiply each element of the matrix \(A\) with \(1.12\) (which represents a 12% increase). This gives us a new matrix \(B\), where \(B = A \times 1.12\).

Find the maximum rate

Locate the maximum rate in the resulting matrix \(B\). The highest number in this matrix represents the maximum room rate for the next season.

Key concepts

Matrix Transformation

Matrix transformation is an essential concept in linear algebra and is widely used in various applications, such as computer graphics, engineering, and economics. In the context of this exercise, the transformation involves manipulating a matrix to analyze changes in room rates over a period.
Matrix multiplication plays a crucial role here, as it allows us to scale or alter the values of a matrix systematically. Given matrix \(A\), which includes the room rates for four hotels, the task involves a scalar multiplication of the matrix by a factor representing a potential price increase.

• Start by identifying matrix \(A\) which describes the current rates for double and family occupancies at different hotels.
• Apply a transformation by multiplying this matrix with a scalar (1.12) to simulate a percentage increase across all rates, resulting in a new matrix \(B\).
• Each element of the original matrix is scaled individually, meaning matrix transformation affects all rates uniformly.

Recognizing and utilizing matrix transformations allows for efficient calculations and analyses, especially in scalable data scenarios like price adjustments or quantitative data modeling.

Room Rates Calculation

The calculation of room rates involves determining how these rates can change due to various factors such as seasonality, demand, and potential price increases. Here, we focus on using matrix operations to predict how room rates might change in the future.
By representing room rates with a matrix and applying a computed transformation, students can easily see how individual room categories across different hotels react to price shifts.

• The matrix \(A\) encapsulates all initial room rates, with each row or column representing different rooms or hotels.
• By multiplying each entry by a factor (in this exercise, 1.12), the resulting matrix \(B\) shows the expected future rates accounting for a specific increase.
• This method does not only enhance the room rate calculations but also provides clarity and visualization of how varying rates are affected uniformly across the board.

This approach simplifies the calculation process and makes it more intuitive for students to predict budget changes in settings like hotel management and hospitality economics.

Percentage Increase Analysis

Analyzing percentage increases involves evaluating how much a certain value grows relative to its original amount. It's a common analysis used in finance, economics, and everyday scenarios like determining cost adjustments.
To evaluate the percentage increase in this exercise, it is essential to understand the steps of converting a percentage to a multiplicative factor.

• Identify the current values that require adjustment—here, the room rates encapsulated within a matrix form.
• Determine the percentage increase, such as 12%. To translate this into a mathematical operation, convert the percentage to a decimal and add 1, yielding 1.12 as the factor.
• Apply this factor to the entire matrix to reflect a new reality encompassing the maximum possible rate hikes.

Percentage increase analysis using matrix operations demonstrates systematic financial forecasting. It offers a practical approach to handle bulk calculations while maintaining accuracy and simplicity.