Let P(t) be the population (in millions) of a certain city t...

Let $P(t)$ be the population (in millions) of a certain city $t$ years after 2015, and suppose that $P(t)$ satisfies the differential equation $$P^{\prime}(t)=.03 P(t), P(0)=4$$ (a) Use the differential equation to determine how fast the population is growing when it reaches 5 million people. (b) Use the differential equation to determine the population size when it is growing at the rate of 400,000 people per year. (c) Find a formula for $P(t).$

Let $P(t)$ be the population (in millions) of a certain city $t$ years after 2015, and suppose that $P(t)$ satisfies the differential equation
$$P^{\prime}(t)=.03 P(t), P(0)=4$$
(a) Use the differential equation to determine how fast the population is growing when it reaches 5 million people.
(b) Use the differential equation to determine the population size when it is growing at the rate of 400,000 people per year.
(c) Find a formula for $P(t).$

Key Concepts

Exponential Growth Model

This model describes a situation where the rate of change of a quantity is directly proportional to the current amount of that quantity, typically expressed as dP/dt = kP. It is commonly used in contexts such as population growth, where under ideal conditions, increases occur at a rate proportional to the existing population.

Separable Differential Equations

A separable differential equation is one in which the variables can be separated on opposite sides of the equation, allowing the problem to be reduced to integrating each side independently. This technique is essential for solving simple differential equations that model exponential growth.

Initial Value Problems

An initial value problem involves a differential equation along with an initial condition, which specifies the value of the unknown function at a particular point in time. The initial condition is crucial because it is used to determine the specific solution from the general solution by fixing the constant of integration.

Interpreting the Derivative as a Rate of Change

Within the context of population models or similar scenarios, the derivative represents the instantaneous rate at which the quantity is changing over time. Understanding this concept allows one to relate the mathematical formulation of the derivative to real-world measures, such as how quickly a population is growing or declining at a particular moment.

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