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Chapter 6: Problem 11

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 population of a town with a 2010 population of 90,000 grows at a rate of \(2.4 \% /\) yr. In what year will the population double its initial value (to \(180,000) ?\)

Short Answer

Expert verified
Answer: 2039

Step by step solution

01

Convert the percentage growth rate to a decimal

To convert the given annual growth rate of \(2.4\%\) to a decimal, divide by \(100\): \(r = \frac{2.4}{100} = 0.024\)
02

Write the exponential growth function

With the initial population \(N_0 = 90,000\) and the growth rate \(r = 0.024\), we can write the exponential growth function for the town's population: \(N(t) = 90,000 * (1+0.024)^t\)
03

Set up the equation to solve for t

We want to find the time \(t\) when the population doubles to \(180,000\), so we will set \(N(t) = 180,000\): \(180,000 = 90,000 * (1+0.024)^t\)
04

Solve for t

Divide both sides of the equation by \(90,000\): \(\frac{180,000}{90,000} = (1+0.024)^t\) \(2 = (1.024)^t\) Now, we will use the logarithm to solve for t: \(log(2) = log((1.024)^t)\) \(\Rightarrow t * log(1.024) = log(2)\) \(\Rightarrow t = \frac{log(2)}{log(1.024)}\) Calculate the value of t: \(t \approx 29.02\)
05

Calculate the year when the population doubles

We found that the population will double in approximately \(29.02\) years. Since the initial year is \(2010\), we will add \(29\) years to find the year when the population doubles: \(2010 + 29 = 2039\) Thus, the population will double its initial value to \(180,000\) in the year \(\boxed{2039}\).

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Key Concepts

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

Population Growth
Population growth is the increase in the number of individuals in a population, like the residents of a town or city. It's a concept often observed in biological entities, but it's just as applicable to human populations. When we talk about the population growth of a town, we are referring to the change in the number of its inhabitants over time.

Various factors affect population growth, such as:
  • Birth rates
  • Death rates
  • Migration trends
In this exercise, we focus on exponential growth, an important type of population growth where the increase is proportional to the current population. That means the larger the population, the faster it grows, continually accelerating over time.
Exponential Function
An exponential function is a mathematical expression that models situations where growth or decay accelerates rapidly. In the case of exponential growth, the function describes how a quantity (like population) increases over time. The general form of an exponential growth function is:
  • Expression: \(N(t) = N_0 \times (1 + r)^t\)
  • \(N(t)\) is the population at time \(t\)
  • \(N_0\) is the initial population size
  • \(r\) is the growth rate (as a decimal)
  • \(t\) is the time that has passed
In our specific example, the town starts with a population of 90,000 in the year 2010, and grows according to this formula. The rate is given as a percentage, which is converted to a decimal to work with this equation.
Logarithms
Logarithms are mathematical tools used to solve equations involving exponents. They help us "undo" exponentiation, allowing us to find unknown exponent values when we know the base result. In simpler terms, logarithms are the opposite of exponentiation.

When we solve for time \(t\) in an exponential growth equation like \(2 = (1.024)^t\), we use logarithms to isolate \(t\):
  • Take the logarithm of both sides: \(\log(2) = \log((1.024)^t)\)
  • Utilize the power rule of logarithms, \(\log(a^b) = b\cdot\log(a)\): \(t \cdot \log(1.024) = \log(2)\)
  • Solve for \(t\): \(t = \frac{\log(2)}{\log(1.024)}\)
Using logarithms simplifies this process and allows us to find the precise time it takes for the population to reach double its original size.
Growth Rate Calculation
Calculating the growth rate involves determining how quickly a population increases. In exponential growth, this rate remains constant, creating a compounded effect.

To calculate the growth rate, we convert the percentage into a decimal by dividing by 100. For example, a growth rate of 2.4% becomes \(0.024\).
The exponential growth model, \(N(t) = N_0 \times (1 + r)^t\), relies on:
  • A known initial population, \(N_0\)
  • A growth rate, \(r\), dictating how much the population increases annually
  • Time \(t\) to predict future population size
This exercise involves using the initial population of 90,000, a growth rate of 2.4%, and finding the time when the population doubles. By solving \(t\), we converted log equations to find the time, giving us precise predictions about future growth.

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