the population of a city is modeled by the equation (p(t)=230,000 + 700t^{2}), where (t) is the number of…
the population of a city is modeled by the equation (p(t)=230,000 + 700t^{2}), where (t) is the number of years since 2000. find the rate of change of the population after 5 years. the population after 5 years is ? by ?
Answer
Explanation:
Step1: Find the derivative of the population function
The derivative of $P(t)=230000 + 700t^{2}$ with respect to $t$ using the power - rule. The derivative of a constant is 0 and the derivative of $at^{n}$ is $nat^{n - 1}$. So $P^\prime(t)=\frac{d}{dt}(230000)+\frac{d}{dt}(700t^{2})=0 + 1400t=1400t$.
Step2: Evaluate the derivative at $t = 5$
Substitute $t = 5$ into $P^\prime(t)$. We get $P^\prime(5)=1400\times5$. $P^\prime(5)=7000$.
Answer:
7000