Problem 5 - College Algebra with Corequisite Support

Question

For the following exercises, identify whether the statement represents an exponential function. Explain. A population of bacteria decreases by a factor of \(\frac{1}{8}\) every 24 hours.

Step-by-Step Solution

Answer

Yes, it represents an exponential decay function.

Step 1: Identify the Problem

We need to determine if the statement about the bacteria decreasing represents an exponential function. An exponential function typically involves a quantity changing at a constant rate per a unit of time.

Step 2: Understand Key Characteristics of Exponential Functions

Exponential functions are characterized by a constant multiplicative rate of change. If the rate of change, or growth/decay factor, is consistent across equal intervals, then the function is exponential.

Step 3: Analyze the Given Statement

The statement says that the population of bacteria decreases by a factor of \(\frac{1}{8}\) every 24 hours. This means the population is multiplied by a constant (\(\frac{1}{8}\)) every time period (24 hours), which is a consistent multiplicative change.

Step 4: Conclude Whether the Statement Represents an Exponential Function

Since the rate of change is consistently a multiplication by \(\frac{1}{8}\) every 24 hours, this is a classic example of an exponential decay function. In mathematical terms, it can be represented as \( P(t) = P_0 \cdot \left(\frac{1}{8}\right)^{\frac{t}{24}} \), where \( P_0 \) is the initial population and \( t \) is the time in hours.

Key Concepts

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. In our exercise, the bacteria population decreases by a factor of \( \frac{1}{8} \) every 24 hours. This is a classic example of exponential decay. Instead of subtracting the same number over time, the population shrinks through consistent division by \( \frac{1}{8} \).Think of exponential decay as a rapid shrinking process. It’s common in situations like radioactive decay or depreciation of commodities. The formula representing exponential decay typically looks like \( P(t) = P_0 \cdot b^{\frac{t}{T}} \). Here, \( P_0 \) is the initial amount, \( b \) (less than 1) is the decay factor, and \( T \) is the time interval for each decay cycle.

• Decay is multiplicative, not additive.
• Decimal factors < 1 show reduction in quantity.
• Useful for modeling natural or financial processes.

This shrinking isn't linear or fixed; it reflects real-world processes where reductions become smaller as time progresses.

Multiplicative Rate of Change

In exponential functions, the rate of change is multiplicative. This means that instead of adding or subtracting a fixed amount, you multiply the current amount by a constant factor. This constant factor in exponential decay is usually less than 1, indicating a reduction rather than growth.For example, a factor of \( \frac{1}{8} \) implies that every interval, the amount reduces to one-eighth of its previous value. This consistent multiplication is key to exponential functions, distinguishing them from linear functions.\( P(t) = P_0 \cdot \left(\frac{1}{8}\right)^{\frac{t}{24}} \):- \( P_0 \) is the initial population- \( \frac{1}{8} \) is the multiplicative decay factor- \( \frac{t}{24} \) indicates the interval in terms of hoursUnderstanding the multiplicative rate helps you predict behaviors over numerous intervals, providing a powerful tool for anticipating future values in a diminishing sequence.

Constant Rate

A constant rate in the context of exponential functions refers to the unchanging factor by which a quantity changes over time. In the scenario of exponential decay, the rate of change remains steady, such as a constant factor of \( \frac{1}{8} \) every 24 hours for our bacteria population example.This constancy is crucial because it allows for predictable modeling of processes. Regardless of the quantity's size, it will decrease consistently by the same multiplicative factor.- Always the same per interval.- Facilitates straightforward predictions.- Key for distinguishing exponential functions from others.When the rate is constant, it guarantees that across identical time frames, the proportional change remains the same, whether you're observing a bacteria colony or tracking the half-life of a substance. Such regularity in change is what makes exponential functions both powerful and reliable for various applications.