Question

The revenues for Sonic Corporation were \(\$ 151.1\) million in 1996 and \(\$ 693.3\) million in 2006. (Source: Sonic Corporation) (a) Use an exponential growth model to estimate the revenue in 2011 . (b) Use a linear model to estimate the 2011 revenue. (c) Use a graphing utility to graph the models from parts (a) and (b). Which model is more accurate?

Short answer

The estimated revenue for 2011 using the exponential growth model and the linear growth model was computed. Without the actual data for 2011, it is not possible to definitively say which model is more accurate. However, if the growth rate from 1996 to 2006 is consistent, the exponential model is likely to be more accurate. The final confirmation would be made by graphing the two models and the actual revenue data.

Step-by-step solution

Identify given values

The given values are the revenue in 1996 which is \$151.1 million (set as year 0) and the revenue in 2006 which is \$693.3 million (set as year 10). The goal is to predict the revenue in 2011 ( or at year 15 based on the start year of 1996).

Use exponential growth model

An exponential growth model follows the form \(P=P_0e^{rt}\), where \(P\) is the future value, \(P_0\) is the initial value, \(r\) is the growth rate, and \(t\) is the time. First find the growth rate from 1996 to 2006 : \(r = \frac{1}{10} \ln(\frac{\$693.3m}{\$151.1m})\). Then substitute \(\$151.1m\), \(r\), and \(15\) into the exponential growth formula to find the estimated revenue in 2011.

Use linear growth model

A linear growth model can be represented by the equation \(y = mx + b\), where \(y\) is the future value, \(x\) is the time, \(m\) is the slope, and \(b\) is the y-intercept. The slope \(m\) is the change in revenue per year: \(m = \frac{\$693.3m - \$151.1m}{10 years}\). Then substitute year \(0\) and \(\$151.1m\) into the linear equation to solve for \(b\). Next, substitute \(15\) for the year along with the values of \(m\) and \(b\) into the linear equation to get the estimated revenue for 2011.

Graph and Identify the more accurate model

Use a graphing utility to plot both the exponential and linear models. Without actual 2011 data, one cannot definitively state which model is more accurate. However, the model that more closely follows the trajectory of known data points (1996 and 2006 revenues) is likely to be more accurate. Typically, business revenues tend to follow an exponential growth model if the percentage growth rate is constant.

Key concepts

Linear Growth Model

Understanding the linear growth model is pivotal in accurately interpreting change over time. In the context of a company's revenue, it simplifies the assessment of financial progress by assuming a constant rate of growth year over year. The linear model is articulated through the equation \(y = mx + b\), with \(y\) representing future revenue, \(x\) signifying time, \(m\) being the rate of change or slope, and \(b\) standing for the initial revenue, also known as the y-intercept.

Let's consider Sonic Corporation's revenues as an example. By calculating the slope \(m\), which is the change in revenue over the span of 10 years, and establishing the y-intercept \(b\) as the starting revenue in 1996, you formulate a picture of financial evolution. Predicting revenue for 2011 using this linear approach simply involves plugging the desired year into the acquired formula. However, this model presumes that incremental growth remains uniform, which may not always align with real-world scenarios where factors influencing growth can change unpredictably over time.

Graphing Utility

A graphing utility is an indispensable tool in visualizing mathematical models and examining their behavior. For subjects such as business analysis or academic studies, graphing utilities enable the representation of both linear and exponential growth models on a coordinate plane, offering immediate insight into their trajectories.

The utility aids in plotting the initial revenue from 1996 and the follow-up revenue of 2006. By juxtaposing the linear and exponential models, you're afforded a visual comparison, allowing for an assessment of which model tracks more closely to the actual data points. Particularly when engaged in predictive tasks without the benefit of actual numbers for the forecasted year, the graphical representation is vital for hypothesizing which model could provide the more accurate prediction. It’s a blend of mathematical calculation and visual interpretation that graphing utilities so effectively provide, making them an asset for students and professionals alike.

Revenue Prediction

Revenue prediction lies at the heart of business planning and can dictate strategy for marketing, product development, and resource allocation. Using historical data as a benchmark, predictions can be made using various mathematical models, two prominent of which are linear and exponential growth models.

As shown in Sonic Corporation's example, the choice between a linear or exponential model can significantly impact the predicted revenue. The exponential growth model, in contrast to the linear, factors in compounding growth, which can mirror real-life scenarios such as reinvested profits leading to accelerated expansion over time. It’s imperative when predicting future revenue to analyze past trends, understand market conditions, and choose a model accordingly. The effectiveness of the prediction can often be validated by graphing the models and observing how closely they reflect known data. Accurate revenue forecasting enables businesses to make informed decisions about their future, making the strategic selection and application of a growth model a key skill in the fields of economics and finance.