Problem 13 - Algebra A Combined Function

Question

Solve. Unless noted otherwise, round answers to the nearest whole. Suppose a city with population 500,000 has been growing at a rate of \(3 \%\) per year. If this rate continues, find the population of this city in 12 years.

Step-by-Step Solution

Answer

The population will be approximately 712,880 in 12 years.

Step 1: Understand the Problem

We need to calculate the future population of the city after 12 years given its current population and annual growth rate.

Step 2: Identify the Formula

We will use the formula to calculate future value with continuous annual growth: \( P(t) = P_0 (1 + r)^t \), where \( P(t) \) is the future population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the number of years.

Step 3: Substitute Known Values

For this problem, substitute \( P_0 = 500,000 \), \( r = 0.03 \), and \( t = 12 \) into the formula: \[ P(12) = 500,000 (1 + 0.03)^{12} \]

Step 4: Calculate the Growth Factor

Compute \( (1 + 0.03)^{12} \): \[ 1.03^{12} \approx 1.42576 \]

Step 5: Find the Future Population

Multiply the initial population by the computed growth factor: \[ P(12) = 500,000 \times 1.42576 \approx 712,880 \]

Step 6: Round to the Nearest Whole Number

The calculated population is 712,880. When rounded to the nearest whole number, the population is 712,880.

Key Concepts

Population Modeling

Population modeling is a way to predict how the number of individuals in a population will change over time. It commonly involves mathematical formulas and statistical data to forecast future growth. In scenarios involving a constant rate of growth, exponential models are often used, making it easier to estimate future populations.

This concept applies not just to cities, but also to understanding how animal populations or even the spread of viruses might evolve.

• Start with a baseline, or current population, like 500,000 in our example.
• Establish the growth rate, such as 3% annually, which remains constant over time.

A critical aspect of population modeling is choosing the right method for your data—for instance, using exponential models for steady growth rates. These models take into account compounding increases, providing a more accurate picture of what the future might hold.

Future Value Calculation

When we're discussing future value calculations in population modeling, we focus on predicting how large a number (like a population) will grow given a consistent rate. This calculation helps provide an estimate of future scenarios based on current data, which is crucial for urban planning, resource allocation, and more.

The formula we use for future value calculation in this scenario is:

• \[P(t) = P_0 (1 + r)^t\]
• \(P(t)\) is the future population size.
• \(P_0\) is the initial population.
• \(r\) is the growth rate.
• \(t\) is the time in years.

This powerful formula allows you to enter the known parameters, such as a 3% growth over 12 years, and solve for the future population. Using formulas like these enables city planners and researchers to visualize what their communities might need in the future.

Annual Growth Rate

The annual growth rate is a key factor in exponential growth models and future value calculations. It tells you how much the initial quantity, such as a population, grows each year. In the example, the city population grows by 3% each year.

• Annual growth rate is represented by \(r\) in our formula.
• It is expressed as a decimal in calculations, like 0.03 for 3%.

The annual growth rate is a crucial measure because it shows the potential for growth or decline over time. This measure helps communicate how quickly a population or investment is increasing and identifies whether that growth is sustainable. By understanding this rate, stakeholders can make informed decisions about the best ways to manage resources and handle future challenges. Calculating with this growth rate gives you a systematic approach to predict future trends effectively.