Sales of New Products Dodds ^29 created a mathematical model...

Sales of New Products Dodds $^{29}$ created a mathematical model that related the sales $S$ of color television sets to the number $t$ of years after introduction and given by $$ S(t)=418,000 \frac{e^{-0.842 t}}{\left(1+154 e^{-0.842 t}\right)^{2}} $$ Determine the time for which $S(t)$ reaches its maximum.

Sales of New Products Dodds $^{29}$ created a mathematical model that related the sales $S$ of color television sets to the number $t$ of years after introduction and given by
$$
S(t)=418,000 \frac{e^{-0.842 t}}{\left(1+154 e^{-0.842 t}\right)^{2}}
$$
Determine the time for which $S(t)$ reaches its maximum.

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Key Concepts

Exponential Functions

Exponential functions are characterized by the presence of variables in the exponent, and they are frequently used to model processes that involve growth or decay. Their properties are particularly important in modeling real-world phenomena, where understanding the rate and nature of change is crucial.

Optimization

Optimization involves the process of finding the maximum or minimum values of a function, which is crucial in various fields including economics and engineering. Using techniques such as setting the derivative equal to zero and analyzing the behavior of the function at these points helps determine the optimal solution.

Critical Points

Critical points are the values of the variable where the derivative of a function is zero or does not exist. These points are significant because they indicate possible locations for local extrema (maximum or minimum values) of the function.

Differentiation

Differentiation is a fundamental calculus tool used to calculate the rate at which a function changes with respect to its variable. It is essential for finding where a function’s slope is zero or undefined, which in turn helps identify potential maximum or minimum points in optimization problems.

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