Question

The management of a company is considering three possible models for predicting the company's profits from 2003 through 2008 . Model I gives the expected annual profits if the current trends continue. Models II and III give the expected annual profits for various combinations of increased labor and energy costs. In each model, \(p\) is the profit (in billions of dollars) and \(t=0\) corresponds to 2003 . Model I: \(\quad p=0.03 t^{2}-0.01 t+3.39\) Model II: \(\quad p=0.08 t+3.36\) Model III: \(p=-0.07 t^{2}+0.05 t+3.38\) (a) Use a graphing utility to graph all three models in the same viewing window. (b) For which models are profits increasing during the interval from 2003 through 2008 ? (c) Which model is the most optimistic? Which is the most pessimistic? Which model would you choose? Explain.

Short answer

Model I and Model II are showing increasing profits from 2003 to 2008. Model II is the most optimistic and Model III is the most pessimistic. The choice of model depends on belief in the underlying assumptions of each model.

Step-by-step solution

Graph the Models

The first step to solving this problem is to graph all three models in the same viewing window so we can visualize and compare them. The graph is not provided here but can be plotted in any graph plotting software or calculator that supports function plotting.

Identify Increasing Profits

We are to determine which models show an increase in profits from 2003 to 2008. In terms of the given functions, this corresponds to an increase when \(t\) goes from 0 to 5 (since \(t=0\) is 2003 and \(t=5\) is 2008).\n\nFor Model I, \(\frac{d}{dt}(0.03t^{2} - 0.01t + 3.39)= 0.06t - 0.01\). The derivative is positive for \(t\) in [0,5], so profits increase.\n\nFor Model II, \(\frac{d}{dt}(0.08t + 3.36)=0.08\), which is always positive. So the profits are always increasing.\n\nFor Model III, \(\frac{d}{dt}(-0.07t^{2} + 0.05t + 3.38)= -0.14t + 0.05\), which is negative for \(t\) in [0,5]. Hence, the profits decrease.

Compare Models

The most optimistic model is the one that predicts the highest profits. Model II has a constant growth rate as it is a linear function, while the other two are quadratic. By comparing the function values at \(t=5\) (2008), we see that Model II provides the most optimistic view with the highest profit value. Model III projects a decrease in profits over time, so it is the most pessimistic. Which model to choose will depend on the belief in the underlying assumptions of each, which are not given here. However, a model showing consistent growth (Model II) might be favored if the increase in labor and energy costs, which this model represents, is believed to not have a major effect on profit.

Key concepts

Graphing Functions

Graphing functions is a crucial early step when analyzing mathematical models, as it provides a visual representation of how profits are projected to change over time in each scenario. In the exercise, we graph three different models to compare their predictions for company profits from 2003 to 2008. Graphing involves plotting each function on a set of axes where the x-axis typically represents time, and the y-axis represents profits. By graphing all models in the same window, it becomes easier to visually compare the trajectories of each. Using software or a calculator capable of plotting functions, you input each equation: Model I, Model II, and Model III. The visual differences, such as whether a curve slopes upward or downward or whether it is a straight line, can help quickly identify which model shows increasing or decreasing profits. This graphical analysis is key in determining the best model for predicting profits, making the interpretation of mathematical expressions more intuitive and accessible.

Derivative Analysis

Derivative analysis helps us understand how the rate of a function's change can signify increasing or decreasing trends. When analyzing profit models, as in the exercise, derivatives provide a tool to identify whether profits are growing over a specific interval. To find whether the profits are increasing, we calculate the derivative for each model. For a function \(p(t)\), the derivative \(\frac{d}{dt}p(t)\) tells us the rate of change of profit concerning time:

• Model I: \(\frac{d}{dt}(0.03t^{2} - 0.01t + 3.39) = 0.06t - 0.01\). Since this expression is positive in the interval \([0,5]\), profits are increasing.
• Model II: \(\frac{d}{dt}(0.08t + 3.36) = 0.08\), which is always positive, indicating profits consistently rise.
• Model III: \(\frac{d}{dt}(-0.07t^{2} + 0.05t + 3.38) = -0.14t + 0.05\), which turns out negative over the interval, meaning profits decrease.

Understanding the positivity or negativity of these derivatives over the time interval allows us to predict the trend, such as constant growth, acceleration, or decline, which is pivotal in making informed decisions based on the models provided.

Quadratic and Linear Models

The exercise involves both quadratic and linear models to predict company profits, each with distinct mathematical characteristics:

• **Quadratic Models:** These models, such as Model I and Model III, are represented by quadratic equations of the form \(at^2 + bt + c\). They have parabolic graphs which can either be concave up or down. Model I's function opens upwards, suggesting the potential for increasing profits. Conversely, Model III opens downward, typically indicating decreasing profits over time.Linear models like Model II use the equation form \(at + b\) and graph as straight lines. Unlike quadratic models, they exhibit a constant rate of change.
• **Linear Models:** Model II is an example, using \(0.08t + 3.36\) to project steady profit growth. The linear model’s simplicity stems from its constant slope, easy interpretation, and predictability over specified intervals.

Each type of model suits different assumptions and scenarios. A quadratic model can capture more complex, non-linear patterns of growth or decline. In contrast, the linear model’s consistent rate make them straightforward and reliable for steady increases. Preferences for one model over another depend on whether the assumptions about growth rates align with expected realities.