Exponential Functions

Question

Research scientists released 500 turtles into a wetland with hopes of repopulating the habitat. After 5 years, the reasearch team estimated that the population had grown to 929 turtles. Predict the population of turtles after 10 years. Round your answer to the nearest integer.

Answer 1

This is an exponential growth word problem. Let's first look at the Exponential Growth Formula.

Step 0: Exponential Growth Formula

P(t) = Ae^kt

'A' is the initial population, 'e' is the Euler constant, 'k' is the growth constant, 't' is the time, and 'P(t)' is the population at time 't'.

Step 1: Definitions

We are told that initially, "research scientists released 500 turtles into a wetland". That means we can define 'A', the initial population.

A = 500

We are also told that "After 5 years, the research team estimated that the population had grown to 929 turtles." That means we can define 't' and 'P(t)'.

t = 5, P(5) = 929

Step 2: Substitute Into Population Equation

P(t) = Ae^kt

P(5) = 500e^5k

929 = 500e^5k

Note that the only variable that we don't know is 'k', the growth constant. Let's use algebra to solve for it.

Step 3: Solving For the Growth Constant

Let's divide both sides of the equation by 500.

929 / 500 = 500 e^5k / 500

929 / 500 = e^5k

Let's take the natural log of both sides of the equation.

ln(929 / 500) = ln(e^5k)

ln(929 / 500) = 5k

Let's divide both sides of the equation by 5.

ln(929 / 500) / 5 = 5k / 5

ln(929 / 500) / 5 = k

Putting this in a calculator, we get a value for 'k'.

k = 0.1239

Step 4: Population Function

Now that we have k, we can write a complete population function for the turtles.

P(t) = Ae^kt

P(t) = 500e^0.1239t

Step 5: Population Prediction

We are asked to "predict the population of turtles after 10 years." This means that we know a value for the time 't'.

t = 10

Let's substitute t = 10 into our new population equation.

P(t) = 500e^0.1239t

P(10) = 500e^{0.1239 * 10}

Putting this in a calculator, we get a value for P(10).

P(10) = 1726.082

There are approximately 1,726 turtles after 10 years.