Based on data obtained by polling automobile buyers, the...

Based on data obtained by polling automobile buyers, the number of subscribers of satellite radios is expected to grow at the rate of $$ r(t)=-0.375 t^{2}+2.1 t+2.45 \quad(0 \leq t \leq 5) $$ million subscribers/year between $2003(t=0)$ and 2008 $(t=5)$. The number of satellite radio subscribers at the beginning of 2003 was $1.5$ million. a. Find an expression giving the number of satellite radio subscribers in year $t(0 \leq t \leq 5)$. b. Based on this model, what was the number of satellite radio subscribers in 2008 ?

Based on data obtained by polling automobile buyers, the number of subscribers of satellite radios is expected to grow at the rate of
$$
r(t)=-0.375 t^{2}+2.1 t+2.45 \quad(0 \leq t \leq 5)
$$
million subscribers/year between $2003(t=0)$ and 2008 $(t=5)$. The number of satellite radio subscribers at the beginning of 2003 was $1.5$ million.
a. Find an expression giving the number of satellite radio subscribers in year $t(0 \leq t \leq 5)$.
b. Based on this model, what was the number of satellite radio subscribers in 2008 ?

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Key Concepts

Rate of Change and Integration

This concept refers to understanding how a quantity changes over time and using integration to accumulate these changes. The rate of growth given by a function is integrated to obtain the total accumulation of a quantity, such as the total number of subscribers in this context. Integration serves to 'sum up' the effect of the instantaneous rate over a period, transforming a derivative or rate function back into the original accumulated function.

Initial Value Problems

An initial value problem combines a differential equation with a given starting value. After integrating the rate function, the constant of integration is determined using the initial condition provided. This concept ensures that the particular solution of the integrated function accurately reflects the real situation at the start of the measured time period.

Polynomial Integration

This concept involves integrating functions that are expressed as polynomials. Each term of a polynomial is integrated separately using standard rules (e.g., increasing the exponent by one and dividing by the new exponent). This method is straightforward and is particularly applicable to cases where the rate function is a quadratic or other polynomial form, as seen in the given modeling scenario.

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