Problem 75 - College Algebra Essentials

Question

Make Sense? In Exercises \(73-76\), determine whether each statement makes sense or does not make sense, and explain your reasoning. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

Step-by-Step Solution

Answer

Yes, the statement makes sense because in the context of exponential functions, a negative growth rate (\(k\)) indicates a decreasing or declining quantity over time, which aligns with the description of a declining population.

Step 1: Understand the problem

Exponential functions are generally written as \(y(t) = a \cdot e^{kt}\), where \(y(t)\) represents the quantity being measured, \(a\) is the initial quantity, \(t\) is time, \(k\) is the growth rate, and \(e\) is Euler's number, a mathematical constant. When \(k > 0\), we're modeling growth; when \(k < 0\), we're modeling decline. The phrase describes using an exponential function to model Russia's declining population, suggesting a negative growth rate. This matches the use of \(k < 0\) for a decline in exponential functions.

Step 2: Explain your reasoning

Given the description is about a declining population, it would be modeled with an exponential function having a negative growth rate. The negative growth rate implies a decline or decrease over time. So, the statement, 'When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.' does make sense.

Key Concepts

Growth Rate

The growth rate is a key component in exponential functions, which allows us to understand how a quantity changes over time. It's represented by the variable \(k\) in the formula \(y(t) = a \cdot e^{kt}\). Here, \(a\) is the initial amount, \(t\) is the time period, and \(e\) is Euler's number, approximately equal to 2.71828, a constant used in continuous growth calculations.

- A positive growth rate \(k > 0\) indicates that the quantity is increasing over time. This is often referred to as exponential growth.- A negative growth rate \(k < 0\) denotes a decrease or decline, indicating exponential decay.

Understanding whether \(k\) is positive or negative helps us model real-world scenarios accurately. For example, a country with an increasing population would typically have a positive growth rate, while one experiencing a population decrease would have a negative growth rate. This makes the concept of growth rate versatile in modeling various scenarios.

Exponential Decay

Exponential decay is a process where the quantity decreases at a rate proportional to its current value. This is described using an exponential function where the growth rate \(k\) is negative \((k < 0)\).

In the equation \(y(t) = a \cdot e^{kt}\), when \(k\) is negative, it indicates that the quantity is shrinking over time. Some common examples of exponential decay in real life include:

• Radioactive decay, where unstable atoms lose energy over time.
• Depreciation of assets, such as cars losing value the longer they are used.
• Decreasing population sizes, like the example of Russia's declining population.

Exponential decay models are important because they help predict how quickly something will reduce, allowing for better planning and resource management.

Population Modeling

Population modeling uses mathematical functions to simulate the changes in a population over time. One common form is using exponential functions, which can accurately reflect both growth and decay in populations.

When applying an exponential model to populations:

• If the growth rate \(k\) is positive, this indicates a growing population. This could result from economic prosperity, improved healthcare, or high birth rates.
• If \(k\) is negative, it suggests a declining population, possibly due to high emigration rates, lower birth rates, or increased mortality.

Population modeling is crucial for governments and organizations since it influences decisions involving resource allocation, infrastructure planning, and policy making. Using exponential functions provides a clear and effective method to visualize how a population might change over given periods.