Problem 71 - Algebra and Trigonometry

Question

A city was incorporated in 2004 with a population of 35,000. It is expected that the population will increase at a rate of 2% per year. The population \(n\) years after 2004 is given by the sequence $$P_{n}=35,000(1.02)^{n}$$ (a) Find the first five terms of the sequence. (b) Find the population in 2014.

Step-by-Step Solution

Answer

First five terms: 35,000, 35,700, 36,414, 37,142.28, 37,885.13; Population in 2014: 42,653.14.

Step 1: Find the first five terms

\(P_n = 35{,}000(1.02)^n\)

• \(P_0 = 35{,}000(1.02)^0 = 35{,}000\)
• \(P_1 = 35{,}000(1.02)^1 = 35{,}700\)
• \(P_2 = 35{,}000(1.02)^2 = 36{,}414\)
• \(P_3 = 35{,}000(1.02)^3 \approx 37{,}142.28\)
• \(P_4 = 35{,}000(1.02)^4 \approx 37{,}885.13\)

Step 2: Find the population in 2014

2014 is 10 years after 2004, so \(n = 10\):
\(P_{10} = 35{,}000(1.02)^{10} = 35{,}000 \times 1.21899 \approx 42{,}664.65\)

The population in 2014 is approximately \(\boxed{42{,}665}\).

Key Concepts

Population Growth

Population growth is an important aspect that shapes societies. It refers to the increase in the number of individuals in a given population over time. In our example, the city’s population starts at 35,000 in 2004 and grows annually.

Why does population growth matter?

• Opportunity: A growing population can boost local economies by increasing demand for goods and services.
• Challenges: It can also lead to crowded cities and environmental stress if growth isn't well-managed.

The specific growth we see here happens due to a constant growth rate, which we'll understand better under exponential growth. The formula used to calculate the increase over the years is a key tool for predicting future population sizes.

Exponential Growth

Exponential growth is a powerful concept that occurs when a quantity increases by a constant percentage over regular intervals of time. Unlike linear growth, where the population increases by a fixed amount each year, exponential growth increases the population by a fixed percentage.

In the city example, the population grows by 2% each year, meaning this is exponential growth because:

• The population size increases more rapidly each year, as each year's increase is a percentage of the previous year's total.
• This results in a curve on a graph, rather than a straight line. The curve gets steeper over time.

This type of growth can lead to large numbers over time, which makes predicting and planning important for city planners and policymakers.

Sequence Formula

The sequence formula is crucial for calculating each term in our series representing the population over time. In this specific example, the sequence formula is given by \[P_{n} = 35,000 \times (1.02)^{n}\].

• Initial value: 35,000 is the starting population. This is our initial point at year 2004.
• Growth factor: 1.02 represents the 2% growth rate, where 1 corresponds to the existing population, and 0.02 is the growth.
• Exponent \( n \): The exponent \( n \) is the number of years that have passed since 2004. It allows us to calculate what the population will be in any given future year.

By substituting \( n = 0 \), we calculate the population for 2004, and by plugging in \( n = 10 \), we can find the population in 2014. This formula is designed to model how the population changes annually based on a constant growth rate.