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Chapter 6: Problem 13

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. The current population of a town is 50,000 and is growing exponentially. If the population is projected to be 55,000 in 10 years, then what will be the population 20 years from now?

Short Answer

Expert verified
Predict the population 20 years from now using the given data: The initial population is 50,000, and it increases to 55,000 in 10 years. Using the exponential growth function: \(P(t) = 50,000 \cdot e^{\frac{1}{10}\ln \left(\frac{55,000}{50,000} \right)t}\) Calculate the population after 20 years: \(P(20) \approx 60,500\) The population 20 years from now will be approximately 60,500.

Step by step solution

01

Identify the reference point and given data

The reference point \((t=0)\) is the current time, when the population is 50,000. We also know that in 10 years (t=10), the population will be 55,000.
02

Set up the exponential growth equation

The exponential growth function has the form: \(P(t) = P_0 \cdot e^{rt}\) Where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population (50,000 in this case), \(r\) is the exponential growth rate, and \(t\) is the time in years.
03

Find the exponential growth rate

Using the data provided, we can plug in the values for \(t=10\), \(P_0=50,000\), and \(P(t)=55,000\) into the exponential growth equation to find the growth rate, \(r\): \(55,000 = 50,000 \cdot e^{10r}\) Next, we'll solve for \(r\). To do this, divide both sides by 50,000: \(\frac{55,000}{50,000} = e^{10r}\) Now, take the natural logarithm of both sides: \(\ln \left(\frac{55,000}{50,000} \right) = \ln (e^{10r})\) Using the property of logarithms that \(\ln(e^x) = x\): \(10r = \ln \left(\frac{55,000}{50,000} \right)\) Lastly, divide by 10 to get the value of \(r\): \(r = \frac{1}{10}\ln \left(\frac{55,000}{50,000} \right)\)
04

Write the exponential growth function

Now that we have the exponential growth rate, \(r\), we can write the exponential growth function: \(P(t) = 50,000 \cdot e^{\frac{1}{10}\ln \left(\frac{55,000}{50,000} \right)t}\)
05

Calculate the population after 20 years

To find the population in 20 years, plug in \(t=20\) into the exponential growth function and solve for \(P(t)\): \(P(20) = 50,000 \cdot e^{\frac{1}{10}\ln \left(\frac{55,000}{50,000} \right)(20)}\) Evaluate this expression to get the population after 20 years: \(P(20) \approx 60500\) Thus, the population 20 years from now will be approximately 60,500.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reference Point
In any growth model, including exponential growth, identifying a reference point is crucial. The reference point represents the starting point in time, designated as \(t = 0\).
  • In our exercise, the town's current population of 50,000 serves as this reference point.
  • The units of time are given in years.
Recognizing this point sets the stage for understanding how growth is measured. For instance, when the population is said to grow over periods such as 10 years or 20 years, it always refers back to this baseline of 50,000 individuals.
This baseline point is vital for properly setting up mathematical equations that model population changes.
Exponential Growth Rate
The exponential growth rate is a key component in determining how quickly a population or quantity increases over time. In our exercise, this rate is represented by \(r\).
Calculation of this rate involves the initial and projected future populations. Using the data provided initially (\(P_0 = 50,000\)) and at t \(= 10\) years (\(P(t) = 55,000\)), we plug the values into the formula for exponential growth: \(P(t) = P_0 \cdot e^{rt}\).
  • First, solve for \(e^{10r}\) by dividing both sides by the initial population.
  • Then, use the natural logarithm to extract the growth rate.
The calculation ultimately tells us how responsive a population is to change in time, providing insights on forecasts beyond known points.
Population Model
A population model describes how a population changes over time. Exponential growth models are especially helpful when a population increases at a rate proportional to its current size. Our exercise is structured around the equation: \(P(t) = P_0 \cdot e^{rt}\).
  • \(P(t)\) represents the population at time \(t\).
  • \(P_0\) is the starting population. In this case, 50,000.
  • \(e\) is the mathematical constant approximately equal to 2.71828.
  • \(r\) signifies the determined exponential growth rate.
This model is useful because it allows one to predict future population sizes by simply plugging different time values into the equation once \(r\) is known.
Making such predictions helps in understanding and planning for future needs.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is used in mathematics to transform exponential expressions into linear ones, making calculations simpler. In an exponential equation such as found in our task, taking the natural logarithm of both sides helps isolate the exponential component.
  • For instance, converting \(e^{10r}\) to \(10r\) involves taking \(\ln\) of that expression.
  • The property that \(\ln(e^x) = x\) allows us to easily isolate the variable \(r\).
This technique is a fundamental tool in algebra for solving equations involving exponential growth or decay. Knowing how to use natural logarithms allows one to manipulate and solve equations involving e effectively.

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