Problem 69 - Algebra Form and Function

Question

A town's population in thousands in 20 years is given by \(15(1+x)^{20},\) where \(x\) is the growth rate per year. What is the population in 20 years if the growth rate is (a) \(2 \%\) ? (b) \(7 \%\) ? (c) \(-5 \% ?\)

Step-by-Step Solution

Answer

Answer: In 20 years, the population for the respective growth rates will be: (a) 22,260 for a 2% growth rate (b) 58,050 for a 7% growth rate (c) 5,420 for a -5% growth rate

Step 1: Convert the percentages to decimals

We first need to convert the given growth rates (2%, 7%, and -5%) into decimals by dividing each by 100. We get: (a) 2% = 0.02 (b) 7% = 0.07 (c) -5% = -0.05

Step 2: Substitute the growth rates and calculate

Now, we will substitute the values from Step 1 into the equation: \(Population = 15(1+x)^{20}\). We will do this separately for each of the growth rates. (a) Population = \(15(1+0.02)^{20}\) (b) Population = \(15(1+0.07)^{20}\) (c) Population = \(15(1-0.05)^{20}\)

Step 3: Compute the final results

Lastly, we will compute the final population values for each growth rate: (a) Population = \(15(1+0.02)^{20}\) = \(15(1.02)^{20} = 15 \times 1.48 \approx 22.26\) thousands (b) Population = \(15(1+0.07)^{20}\) = \(15(1.07)^{20} = 15 \times 3.87 \approx 58.05\) thousands (c) Population = \(15(1-0.05)^{20}\) = \(15(0.95)^{20} = 15 \times 0.36 \approx 5.42\) thousands In 20 years, the population for the respective growth rates will be: (a) 22,260 for a 2% growth rate (b) 58,050 for a 7% growth rate (c) 5,420 for a -5% growth rate

Key Concepts

Population Modeling

Population modeling is a way to predict how a population will change over time. It uses mathematical formulas to calculate changes in population size. In this exercise, we use a model to estimate how a town's population will grow or shrink over 20 years.
The general formula used here is \( P = 15(1+x)^{20} \),where:

• \( P \) is the population in thousands after 20 years,
• 15 is the initial population in thousands,
• \( x \) is the growth rate per year.

This equation shows exponential growth, which means the rate of change increases over time if \( x \) is positive. If \( x \) is negative, the population decreases. This modeling helps in planning for resources, infrastructure, and other community needs.

Growth Rate Calculation

The growth rate is the percentage at which a population changes over a specific period. In our model, it's represented by \( x \). Adjusting the growth rate changes how quickly the population grows or shrinks.
To determine future population size:

• Identify the growth rate (like 2%, 7%, or -5%).
• Convert this into decimal form so it can be used in calculations.
• Plug the decimal into the formula to find future population given the time span.

For example, with a 2% growth rate represented as 0.02, you plug it into the formula as \( \text{Population} = 15(1+0.02)^{20} \). The result after computation gives the expected population in 20 years. Understanding how to adjust and apply growth rates is critical for accurate future predictions.

Percentage to Decimal Conversion

Direct conversion from percentage to decimal is crucial for performing accurate mathematical calculations. Percentages are a way to express a number as a fraction of 100.
To convert a percentage like 2%, follow these steps:

• Remove the percent sign (%).
• Divide the number by 100.

Thus, 2% becomes 0.02. This small conversion step is vital before plugging values into the exponential growth formula, as mathematical operations require the decimal form.
Applying this conversion:

• 2% = 0.02
• 7% = 0.07
• -5% = -0.05

Accurate conversion ensures proper representation of growth rates in mathematical equations like the one for population growth, leading to high accuracy in projections.