According to a study conducted in 2004, the share of online...

According to a study conducted in 2004, the share of online advertisement, worldwide, as a percentage of the total ad market, is expected to grow at the rate of $$ R(t)=-0.033 t^{2}+0.3428 t+0.07 \quad(0 \leq t \leq 6) $$ percent/year at time $t$ (in years), with $t=0$ corresponding to the beginning of 2000 . The online ad market at the beginning of 2000 was $2.9 \%$ of the total ad market. a. What is the projected online ad market share at any time $t$ ? b. What was the projected online ad market share at the beginning of 2005 ?

According to a study conducted in 2004, the share of online advertisement, worldwide, as a percentage of the total ad market, is expected to grow at the rate of
$$
R(t)=-0.033 t^{2}+0.3428 t+0.07 \quad(0 \leq t \leq 6)
$$
percent/year at time $t$ (in years), with $t=0$ corresponding to the beginning of 2000 . The online ad market at the beginning of 2000 was $2.9 \%$ of the total ad market.
a. What is the projected online ad market share at any time $t$ ?
b. What was the projected online ad market share at the beginning of 2005 ?

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Key Concepts

Modeling with Differential Equations

This concept involves representing the change in a quantity over time using a differential equation. In scenarios like market share growth, the rate of change is provided by a function of time, and modeling this process allows one to understand dynamic behavior over a time period.

Integration to Determine Original Functions

Integrating a given rate function is essential to obtain the original function that describes the accumulated value over time. In this context, integrating the rate of change yields the market share function, which shows how the share increases relative to time.

Application of Initial Conditions

After integrating a rate function, an arbitrary constant of integration is introduced. Using known initial conditions, such as the starting market share value, is crucial to determine this constant and obtain a specific solution that accurately reflects the real-world scenario.

Function Evaluation at Specific Points

Once the integrated function is obtained, evaluating it at a particular time enables the prediction of the state of the system at that time. This technique is used to forecast future values, like the projected market share in a given year.

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