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? $$ R_{1}=7.21+0.26 t+0.02 t^{2}, R_{2}=7.21+0.1 t+0.01 t^{2} $$

Short answer

The model that projects the higher revenue can be identified by comparing the total revenues obtained from both models. The additional total revenue projected by the higher revenue model can be calculated by subtracting the total revenue projected by the model with the lower revenue from the one with higher revenue.

Step-by-step solution

Calculate Annual Revenues for Each Model

The first part is to calculate the revenue predicted by each model for each year from 2007 to 2011. In model formulas, \(t=7\) corresponds to the year 2007, thereby \(t=8, 9, 10, 11\) correspond to the years 2008, 2009, 2010, 2011 respectively. So, plug in these values in both \(R_{1}\) and \(R_{2}\) to find the projected annual revenues.

Sum Up The Revenues

After obtaining the projected revenues for each year from both models, sum up these revenues over the years in both models separately. The sum of revenues from a model over the years gives the total projected revenue from that model for the period.

Compare the Total Revenues

Compare the total revenues obtained from both the models. The model with higher total revenue projects the greater revenue.

Calculate the Difference in Projection

To calculate the additional total revenue projected by the model with higher revenue, subtract the total revenue of the model with lower projection from the model with higher projection. This gives the additional total revenue projected by the model with higher total revenue over the five-year period.

Key concepts

Mathematical Modeling in Economics

Mathematical modeling is a critical tool in economics that provides a simplified representation of complex economic processes. By utilizing mathematical formulas and functions, economists and analysts can predict and analyze the behavior of economic indicators. For instance, in forecasting a company's revenue, modeling involves setting up algebraic equations that account for various factors influencing income, such as market trends, pricing strategies, and historical data.

In our textbook example, models R₁ and R₂ are algebraic representations that estimate a corporation's revenue over specific years. The models are constructed based on the assumption that revenue changes over time as a function of the variable t, which represents the year. Through these models, it is possible to simulate various scenarios and make informed decisions regarding the future of the corporation's finances.

Revenue Estimation

Revenue estimation is a vital part of financial planning for businesses and requires careful analysis and prediction. It typically involves reviewing past performance and considering factors that might affect future income. In our problem, two revenue estimation models are compared to determine which one projects greater income where t corresponds to years.

The method for revenue estimation usually starts with identifying variables that impact revenue and formulating an equation that relates these variables to annual sales. Estimations are not just educated guesses; they are grounded in historical data and growth patterns. By calculating the estimated revenue for each year and adding them up, businesses can plan and allocate resources more efficiently for expected growth or prepare for potential downturns.

Algebraic Equations

Algebraic equations are the foundation of many economic models because they clearly define relationships between variables. In the given exercise, the revenue models R₁ and R₂ are expressed through quadratic equations, and t is the independent variable representing time. The coefficients in these equations – 0.26 and 0.02 for R₁, 0.1 and 0.01 for R₂ – reflect the rates at which revenue is presumed to change yearly.

When using algebraic equations for modeling, it's important to plug in the correct values for the variables and perform accurate calculations. Even minor errors can lead to significant discrepancies in projections, making accuracy paramount in tasks like revenue forecasting. After calculating the individual annual revenues, we sum them to find the total revenue projected by each model.

Calculus Applications

Calculus, particularly differentiation and integration, is widely applied in economics for optimization and area estimation problems. While differentiation helps in finding rates of change and identifying maxima and minima of functions, integration is used for summing up continuous data points over a specific interval, which can be akin to accumulating total revenue over time.

In this textbook case, we essentially perform a discrete sum rather than an integration, due to the problem data given yearly. However, if we had continuous data or wanted to account for variations within the year, we might integrate the revenue function over the desired period to estimate total revenue. Calculus not only helps make more accurate revenue projections but also determines the rate of revenue increase or decrease, which is critical for strategic decision-making.