Question

the population of a city is modeled by the equation $p(t) = 230,000 + 700t^2$, where $t$ is the number of years since 1990. find the rate of change of the population after 5 years.

Explanation:

Step1: Recall the formula for the rate of change (derivative)

The population function is \( P(t) = 230000 + 700t^2 \). The rate of change of the population with respect to time \( t \) is the derivative of \( P(t) \) with respect to \( t \). Using the power rule, if \( y = ax^n \), then \( y' = nax^{n - 1} \), and the derivative of a constant is 0. So, the derivative \( P'(t) \) is calculated as follows:
For the term \( 230000 \) (a constant), its derivative is 0. For the term \( 700t^2 \), using the power rule with \( a = 700 \) and \( n = 2 \), we get \( P'(t)=2\times700t^{2 - 1}=1400t \).

Step2: Substitute \( t = 5 \) into the derivative

We need to find the rate of change after 5 years, so we substitute \( t = 5 \) into \( P'(t) \).
\( P'(5)=1400\times5 \)

Step3: Calculate the value

\( 1400\times5 = 7000 \)

Answer:

The rate of change of the population after 5 years is 7000 people per year.