Intel, a computer chip manufacturer, reported 1265 million...

Intel, a computer chip manufacturer, reported $\$ 1265$ million in total revenue in $1986 .$ In 2005 the total revenue was $\$ 38.8$ billion. (Source: U.S. Securities and Exchange Commission.) Assuming an exponential model, find the growth rate $k$, to four decimal places, and write the revenue function $R,$ with $R(t)$ in billions of dollars. Then predict the company's total revenue for 2012.

Key Concepts

Forecasting Future Values Using the Model

Once the exponential growth model is established and the growth constant k is determined, the model can be used to forecast future values. This involves substituting the desired future time into the model, yielding an estimate based on the continuous compound growth assumption. Such forecasts are valuable tools for predictions in business and economics.

Exponential Growth Model

An exponential growth model is used to represent processes where the quantity increases proportionally to its current value. In such models, the function has the form R(t) = R?e^(kt), where R? is the initial value, k is the growth rate constant, and t is time. This type of model is appropriate for many real-world phenomena where growth compounds continuously over time.

Determining the Growth Rate Using Logarithms

To solve for the growth constant k in an exponential growth model, one can set up an equation based on two known data points and use logarithms to isolate k. This is done by rewriting the equation in the form R(t?) = R?e^(k(t?-t?)), then taking the natural logarithm of both sides to solve for k. This method allows for a precise quantification of the growth rate using the ratio of the future value to the initial value over the elapsed time.

Unit Consistency and Conversion

Maintaining consistency in units is crucial when using mathematical models. For revenue calculations, if the initial value and subsequent values are given in different units (for example, millions versus billions), they must be converted to a common unit before applying the model. This ensures that the model’s parameters correctly reflect the scale of the quantities involved.

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