Problem 554 - Precalculus

Question

The population of a pod of bottlenose dolphins is modeled by the function \(A(t)=8(1.17)^{t},\) where \(t\) is given in years. To the nearest whole number, what will the pod population be after 3 years?

Step-by-Step Solution

Answer

The population after 3 years is 13 dolphins.

Step 1: Understanding the Function

The given function is \(A(t) = 8(1.17)^t\). This represents the dolphin population at a given time \(t\). Here, 8 is the initial population, and 1.17 is the growth factor.

Step 2: Substituting the Time Variable

We are asked to find the population after 3 years. So, substitute \(t = 3\) into the function. This gives us \(A(3) = 8(1.17)^3\).

Step 3: Calculating the Exponent

Compute \((1.17)^3\). This involves multiplying 1.17 by itself three times. \(1.17 \times 1.17 \times 1.17 = 1.580013\).

Step 4: Multiplying by Initial Population

Multiply the result from the previous step by the initial population of 8. So, \(8 \times 1.580013 = 12.640104\).

Step 5: Rounding the Result

Finally, round the number 12.640104 to the nearest whole number to find the population. The nearest whole number is 13.

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a quantity increases or decreases at a rate proportional to its current value. In simpler terms, it means growth or decay happens by a consistent percentage. This type of function is widely used to model growth processes like populations, investments, or any other situation involving repeated percentage changes.

The general form of an exponential function is given by \( f(t) = a(b)^t \), where:

• \(a\) is the initial amount or value
• \(b\) is the base or growth factor
• \(t\) is the time or number of periods

Exponential functions are powerful because they capture how things grow quickly over time. The term "exponential growth" often describes rapid increases due to a constant multiplicative rate, unlike linear growth, which increases at a constant additive rate. Understanding exponential functions is key to interpreting many real-world scenarios, such as population dynamics.

Dolphin Population

When modeling a population, we often turn to functions that can describe how the number of individuals changes over time. For our bottlenose dolphins, the population model is expressed as \( A(t) = 8(1.17)^t \). This function suggests that the pod of dolphins is experiencing growth that can be predicted with time using an exponential model.

In this function:

• The initial population is 8 dolphins. This is the starting point or what we sometimes call the "seed" population. At \( t = 0 \), the population is simply \( A(0) = 8 \).
• The growth factor of 1.17 signifies that the population grows by 17% each year. Each year passed, the population is multiplied by 1.17, pointing to a steady increase.

By evaluating this function at different points in time, such as 3 years, we can estimate population sizes at future points, aiding in conservation and understanding ecological conditions.

Growth Factor

The growth factor is a crucial part of exponential growth models like the one we've used for the dolphin population. It represents how much the quantity grows in each time unit. In our dolphin population model \( A(t) = 8(1.17)^t \), the growth factor is 1.17.

Let's break down exactly what this means:

• A growth factor greater than 1 (like 1.17) implies growth. Specifically, a factor of 1.17 means there is a 17% increase in the population size every year.
• If the growth factor were less than 1, the function would model decay, meaning the population would decrease over time.
• A growth factor of exactly 1 would indicate no change in population size, suggesting a stable population over time.

The way a growth factor impacts a population model is that it dictates the rate at which the population changes in size, which can either speed up growth or slow it down significantly, just as in our dolphin case, where the population size grows substantially over a few years.