Sales The sales S (in thousands of units) of a new CD burner...

Sales The sales $S$ (in thousands of units) of a new CD burner after it has been on the market $t$ years are given by $S=100\left(1-e^{k t}\right) .$ Fifteen thousand units of the new product were sold the first year. (a) Complete the model by solving for $k$ (b) Use a graphing utility to graph the model. (c) Use the graph in part (b) to estimate the number of units sold after 5 years.

Sales The sales $S$ (in thousands of units) of a new CD burner after it has been on the market $t$ years are given by $S=100\left(1-e^{k t}\right) .$ Fifteen thousand units of the new product were sold the first year.
(a) Complete the model by solving for $k$
(b) Use a graphing utility to graph the model.
(c) Use the graph in part (b) to estimate the number of units sold after 5 years.

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Key Concepts

Graphical Analysis

Graphical analysis entails using a graphing utility or other methods to visually represent the behavior of a function. This technique helps in understanding the overall trend and dynamics of the model, verifying solutions, and estimating function values at specific points in time.

Model Interpretation

Model interpretation refers to understanding and conveying what the components of a mathematical model represent in a real-world context. It involves explaining how parameters and the functional form relate to physical quantities and behaviors, such as sales over time in this case.

Exponential Functions

Exponential functions are those in which a constant is multiplied by an exponential term, often expressed as e^(kt). They are used to model situations where the rate of change of a quantity is proportional to the quantity itself. This concept is fundamental to understanding growth and decay processes in various fields.

Parameter Estimation

Parameter estimation involves determining the unknown parameters of a model by using given initial conditions or data points. In the context of an exponential model, this typically requires substituting known values into the equation and solving for the unknown constant, often using logarithmic transformations to isolate the parameter.

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