A study prepared for a Sunbelt town's chamber of commerce...

A study prepared for a Sunbelt town's chamber of commerce projected that the town's population in the next 3 years will grow according to the rule $$ P(t)=50,000+30 t^{3 / 2}+20 t $$ where $P(t)$ denotes the population $t$ months from now. How fast will the population be increasing 9 months and 16 months from now?

A study prepared for a Sunbelt town's chamber of commerce projected that the town's population in the next 3 years will grow according to the rule
$$
P(t)=50,000+30 t^{3 / 2}+20 t
$$
where $P(t)$ denotes the population $t$ months from now. How fast will the population be increasing 9 months and 16 months from now?

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Key Concepts

Population Growth Models

Population growth models are mathematical functions used to represent changes in a population over time. They help in forecasting and analyzing future demographic trends and are commonly used by governments and organizations for planning purposes.

Instantaneous Rate of Change

The instantaneous rate of change measures how rapidly a quantity is changing at a specific moment. This concept is pivotal in calculus and is determined by taking the derivative of the function representing the dynamic process, providing insight into current growth or decline.

Differentiation

Differentiation is the process of calculating the derivative of a function, which quantifies its rate of change. It involves applying rules such as the sum rule, constant multiple rule, and power rule to systematically compute the derivative of composite functions.

Power Rule

The power rule is a basic principle of differentiation used for functions of the form t^n. According to this rule, the derivative of t^n is n multiplied by t raised to the (n-1) power, which simplifies the process of differentiating polynomial expressions, including those with fractional exponents.

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