Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Two models, \(R_{1}\) and \(R_{2}\), are given for revenue (in billions of dollars per year) for a large corporation. Both models are estimates of revenues for 2007 through 2011, with \(t=7\) corresponding to \(2007 .\) Which model is projecting the greater revenue? How much more total revenue does that model project over the five-year period? $$ R_{1}=7.21+0.58 t, R_{2}=7.21+0.45 t $$

Step-by-Step Solution

Answer

Insert the short, quick answer here after completing above steps. Make sure it includes both the model which projects higher revenue and the difference between total revenues predicted by the two models.

Step 1: Evaluate Each Revenue Model Over The Five-Year Period

First, calculate the total revenue obtained from each model over the course of the five years. This entails evaluating each function from \(t=7\) to \(t=11\), then summing those results. So, compute \(R_1(7)\)+\(R_1(8)\)+\(R_1(9)\)+\(R_1(10)\)+\(R_1(11)\) for the first model, and \(R_2(7)\)+\(R_2(8)\)+\(R_2(9)\)+\(R_2(10)\)+\(R_2(11)\) for the second model.

Step 2: Determine The Higher Revenue Model

After calculating the five-year total revenues for each model, compare them to determine which is greater. If \(R_1\) total revenue is greater than \(R_2\) total revenue, \(R_1\) predicts higher revenue, else \(R_2\) predicts higher revenue.

Step 3: Determine The Difference In Revenue

To find how much more revenue the model that predicts higher revenue projects over the five-year period, subtract the total revenue predicted by the model with lower total revenue from the model with higher total revenue.

Key Concepts

Revenue Models

When it comes to understanding how a business will perform financially, revenue models are essential tools. They allow companies to forecast future earnings based on various factors like sales, pricing, and market trends. In our example, the revenue models for a large corporation use linear equations to estimate the company's revenue over a five-year period. The equations take the form of \(R_{1}=7.21+0.58t\) and \(R_{2}=7.21+0.45t\), where \(t\) corresponds to the year, with \(t=7\) representing 2007.

Revenue models can be simple, like the linear models shown here, or complex, incorporating numerous variables and assumptions. The key to a successful revenue model is the accurate prediction of revenues, which can help a business plan for the future. To ensure the reliability of these models, real-world data should be regularly compared to projected figures, and adjustments should be made as necessary.

Linear Functions

Linear functions are a fundamental part of algebra and are used extensively in various fields, including economics and finance, for modeling relationships. They are written in the form \( f(x) = mx + b \) where \( m \) represents the slope, and \( b \) is the y-intercept. In our revenue projection example, the linearity of the function represents a steady rate of change in revenue over time.

Understanding linear functions is crucial for students as these functions predict a direct proportionality between two quantities. When graphed, they produce a straight line, which is visually helpful for comparing different rates of change – in this case, the growth rate of the company's revenue. This simplicity allows for easy calculation and interpretation, which is particularly helpful when making quick business decisions.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world scenarios. In this exercise, we model the projected revenue of a corporation using linear equations. Effective mathematical models can simplify complex systems, such as corporate finances, to improve understanding and forecasting.

It's important to note that the precision of a mathematical model relies heavily on the assumptions it's based on and the data used. Estimations, like the slope of the revenue models \(0.58t\) and \(0.45t\), are derived from historical data and anticipated market conditions. While these models are often a simplification, they are useful tools for research, discussion, and teaching various concepts, like the rate of change in a business context.

Function Evaluation

Function evaluation is a mathematical process where you compute the output of a function given an input value. In the context of our exercise, we evaluate the linear revenue models \(R_{1}\) and \(R_{2}\) by plugging in the values for \(t\) from 7 to 11, which represent the years 2007 through 2011. This involves basic arithmetic—multiplying the slope by the given value of \(t\), then adding the y-intercept.

Function evaluation can be done for different purposes, such as determining a specific point on a graph, comparing outcomes under different scenarios, or as in our case, calculating total revenue over a period. By evaluating both revenue models over five years, we can compare which model projects greater revenue and understand the financial implications for the corporation.