When records were first kept (t=0), the population of a...

When records were first kept $(t=0),$ the population of a rural town was 250 people. During the following years, the population grew at a rate of $P^{\prime}(t)=30(1+\sqrt{t}),$ where $t$ is measured in years. a. What is the population after 20 years? b. Find the population $P(t)$ at any time $t \geq 0.$

When records were first kept $(t=0),$ the population of a rural town was 250 people. During the following years, the population grew at a rate of $P^{\prime}(t)=30(1+\sqrt{t}),$ where $t$ is measured in years.
a. What is the population after 20 years?
b. Find the population $P(t)$ at any time $t \geq 0.$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Differential Equations

Differential equations relate a function to its derivative(s) and are used to model rates of change in many real-world scenarios. In problems involving population growth, the derivative represents the rate at which the population changes, and solving the differential equation helps us find the function that describes the population over time.

Initial Value Problems

An initial value problem is a type of differential equation that comes with a specified value at a certain initial time. This information is used to determine the constant of integration when solving the equation, ensuring that the solution accurately reflects the system's history from its starting point.

Integration

Integration is the process of accumulating changes over time and is used to recover a function given its rate of change. In population modeling, integrating the growth rate function allows us to calculate the total change in population over an interval and to derive the overall population as a function of time.

Population Growth Modeling

Population growth modeling applies differential equations and integration to describe and predict changes in population sizes over time. By using a known initial population and a given growth rate, one can integrate the rate to obtain a function representing the population at any future time.

Transcript

00:03 Okay, so we're going to start with this.

00:06 We have a function that is the population of a rural town.

00:13 It has a population 250, the time equals zero, and its growth rate is this function, 30 times 1 plus the square root of t.

00:23 So what we're going to do is we're going to figure out what is the population after 20 years.

00:28 So figure that out.

00:29 We're going to be finding the integral of p prime from 0 to 20.

00:35 Note, this will tell us how many people are added.

00:39 So at the end, what we need to do to figure out the actual population is add the population at the beginning.

00:47 So we're going to do this.

00:48 This is going to be equal to 30 plus 30 to the part of 1 half, which is then equal to 30 t plus 30 t to the power of 3 over 2 divided by 3 times 2.

01:16 If we simplify that further, we should get 30t plus 20, t3 to the power of 2.

01:32 And we're going to do that from 0 to 20.

01:36 So we get 30 times 20 plus 20 times 20 to the power of 3 over 2 minus 30 times 0 times 0 3 to the part of 2.

01:52 Minus 30 times 0 3 to the part of 2.

01:58 This is equal to 0.

01:59 This is equal to 0 so we can just ignore those.

02:01 And then we have this...

Please give Ace some feedback