Problem 51 - Elementary and Intermediate Algebra

Question

Population Growth. The population of North Rivers is decreasing exponentially according to the formula \(P=3,745(0.93)^{t},\) where \(t\) is measured in years from the present date. Find the population in 6 years, 9 months.

Step-by-Step Solution

Answer

The population in 6 years and 9 months will be approximately 2,299.

Step 1: Understand the Formula

The population of North Rivers decreases exponentially according to the formula \( P = 3,745(0.93)^t \), where \( P \) is the population at time \( t \) years from now. \( 3,745 \) is the initial population, and \( 0.93 \) represents the decay factor.

Step 2: Convert Time to Years

Since we need the population for 6 years and 9 months from now, we need to convert 9 months into years. 9 months is \( 9/12 = 0.75 \) years. Therefore, \( t = 6 + 0.75 = 6.75 \) years.

Step 3: Plug Time into the Formula

Substitute \( t = 6.75 \) into the equation: \[ P = 3,745 \times (0.93)^{6.75} \]

Step 4: Calculate the Exponential Factor

Calculate \( (0.93)^{6.75} \). You may use a calculator for this step: \( (0.93)^{6.75} \approx 0.614125 \).

Step 5: Determine the Population

Multiply the initial population by the calculated exponential factor: \[ P = 3,745 \times 0.614125 \approx 2,299.38 \]. Since population count must be a whole number, round it to 2,299.

Key Concepts

Population Growth

Population growth is a dynamic process that can either be an increase or decrease over time. This growth or decay is commonly described using mathematical models. In the given exercise, population decay is modeled using an exponential function, indicating a consistent rate of decline. Here, we are observing the population of North Rivers decrease over time, starting with an initial population of 3,745 people. It's essential to understand that population changes reflect factors like birth rates, death rates, immigration, and emigration.

For effective planning and development, predicting future population size is crucial. Understanding these concepts helps governments and organizations make informed decisions about resource allocation and management. Analyzing trends using mathematical models offers insight into potential future scenarios based on current or proposed policies.

Exponential Functions

Exponential functions describe processes exhibiting rapid change, where a quantity either grows or decays at a rate proportional to its current value. In this context, the population decay of North Rivers follows an exponential function given by the equation \( P = 3,745(0.93)^t \).

Let's break down the components of this formula:

• \( P \) represents the population at a given time \( t \).
• \( 3,745 \) is the initial population measured at time zero.
• The base \( 0.93 \) is the decay factor, indicating that each year, the population reduces to 93% of what it was the previous year.
• The exponent \( t \) represents the number of years elapsed from the initial measurement.

Exponential decay functions are recognized by their swift progression towards zero, at a rate that’s initially fast and then slows over time. By analyzing this function, one can understand not just present conditions, but also future states, which is crucial for strategic planning.

Time Conversion

Time conversion is a vital skill when dealing with mathematical models that require specific time measurements, especially when time needs to be in uniform units like years or months. In exercises similar to this one, it's important for accuracy in calculations to ensure all time measurements align with the units in the exponential decay formula.

In the given problem, we encountered a situation requiring conversion of 9 months into years. Since there are 12 months in a year, you convert months to years by dividing the number of months by 12. For example, 9 months becomes \( \frac{9}{12} \), equaling 0.75 years. Therefore, to find the total time \( t \), you add the full years to the converted months: \( t = 6 + 0.75 = 6.75 \) years.

This conversion is crucial to solving population growth or decay problems accurately, as the rate changes exponentially over time. Consistency in time units ensures the mathematical model accurately reflects the real-world process being studied.