Problem 14 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A population of beavers is growing exponentially. In June 1993 (our benchmark year when \(t=0\) ) there were 100 beavers. In June \(1994(t=1)\) there were 130 beavers. (a) Write a function \(B(t)\) that gives the number of beavers at time \(t\). (b) What is the percent increase in the beaver population from one year to the next?

Step-by-Step Solution

Answer

The function that gives the number of beavers at time t is \(B(t)=100 \cdot 1.3^t\). The percent increase in the beaver population from one year to the next is 30%.

Step 1: Define the exponential growth function

An exponential growth function could be given as \(B(t) = B_0 \cdot k^t\), where \(B(t)\) is the final population, \(B_0\) is the initial population, \(k\) is the growth multiplier, and \(t\) is the time in years.

Step 2: Solve for \(k\)

From the data given, we know that \(B_0=100\) at \(t=0\) (1993), and \(B(t) = 130\) at \(t=1\) (1994). Substitute these findings into the function and solve for \(k\): \[100 \cdot k^1 = 130\]\Thus, \(k = 1.3\).\]

Step 3: Write the exponential growth function

Substitute the value of \(k\) back into the function, we get the function: \(B(t) = 100 \cdot 1.3^t\). This function gives the number of beavers at time \(t\).

Step 4: Calculate the percent increase

The growth multiplier (k) represents the percent increase from one year to the next. Subtract 1 from \(k\) and multiply it by 100 to get the percentage increase: \( (1.3-1) \cdot 100 = 30% \). Hence, the beaver population increases by 30% from one year to the next.

Key Concepts

Population Growth

Understanding how populations change over time is crucial in fields like ecology, urban planning, and epidemiology. Population growth, particularly exponential growth, is a key concept when it comes to exploring such changes. When we consider how a group of beavers increases in size over time, we can see an example of population growth in action.

Looking at the provided exercise, we see that the number of beavers grew from 100 to 130 in just one year. This increase isn't simply by a constant number of beavers each year, but rather by a percentage of the current population. This type of growth indicates that the larger the population becomes, the more new beavers are added each year, assuming a constant rate of growth. Such a pattern exhibits characteristics of exponential growth, which we will delve into further.

Exponential Function

An exponential function is one of the most important mathematical concepts when dealing with growing populations. Expressed in the form of a math formula, it's usually written as \(B(t) = B_0 \cdot k^t\), where \(B(t)\) represents the population at time \(t\), \(B_0\) is the initial population, and \(k\) is the base of the exponential function, representing the growth rate. The exponent \(t\) stands for time passed.

In our case with the beavers, we identified that with 100 beavers to start with and 130 after the first year, the function can be simplified once we determine the growth multiplier, which turns out to be \(1.3\). This implies the exponential function for our beaver population over time \(t\) is modeled by \(B(t) = 100 \cdot 1.3^t\). With this function, we can calculate the expected population after any number of years.

Rate of Change

The rate of change in the context of exponential population growth refers to how quickly the population is multiplying over time. It is integral to predicting future population sizes and understanding the dynamics of growth. In the context of our exponential function, the rate of change isn't constant; it depends on the current size of the population.

In simpler terms, a population increasing by a fixed ratio (like the beaver population growing at 1.3 times each year) will accelerate in terms of actual numbers added each year. This exponential model naturally reflects many real-world scenarios where resources and space are not limiting. For the beavers' population growth, the rate of change is encapsulated by the multiplier \(1.3\), which reveals how rapidly the population increases from one year to the next.

Percent Increase

Percent increase helps us understand growth in terms of percentages, which is often more intuitive than using multipliers or raw numbers. By converting our multiplier into a percentage, we can state the rate of increase more clearly. If the population grows from 100 to 130 beavers in one year, they have a 30% increase each year, because \((1.3 - 1) \cdot 100 = 30%\).

This percent increase is easy to communicate and understand. It demonstrates how a population isn't just growing; it's growing faster each year by 30% of the population size from the previous year. The percent increase can be particularly compelling when discussing populations that are rapidly expanding, showing the potential for explosive growth over time if the rate continues.