Problem 14 - College Algebra

Question

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. $$ y=300(1-t)^{5} $$

Step-by-Step Solution

Answer

The equation represents neither exponential growth nor decay.

Step 1: Identify the General Form

The general form for an exponential function is \( y = a (b)^t \), where \( a \) is the initial value, \( b \) is the base, and \( t \) is the exponent. Determine if the given equation can be compared to this form.

Step 2: Compare the Equation

The given equation is \( y = 300(1-t)^5 \). Here, \( a = 300 \) and \( b = (1-t) \). This does not match the typical exponential form \( y = a (b)^t \) directly since \( t \) is not the exponent for \( b \).

Step 3: Analyze the Base

In an exponential function, the base \( b \) should be raised to the power of the variable \( t \). Here, \( t \) is a part of the base \( b = 1-t \), instead of being the exponent.

Step 4: Conclusion

Since the variable \( t \) is within the base and not in the exponent, this equation cannot be categorized under standard exponential growth or decay forms. Therefore, it is neither exponential growth nor exponential decay.

Key Concepts

Exponential Growth

Exponential growth occurs when the value of a quantity increases rapidly over time. In mathematical terms, an exponential growth function is generally expressed in the form \( y = a (b)^t \), where the base \( b \) is greater than 1. This indicates that with each increase in variable \( t \), the value of \( y \) doubles or grows by a consistent factor, leading to quick escalation.
In simple words, if something is growing exponentially, it means it's expanding increasingly faster with every tick on the clock. This pattern is common in populations, investments, and certain natural phenomena.
### Recognizing Exponential Growth:

• The base \( b \) in the equation must be greater than 1.
• The exponent \( t \) must be applied directly to the base \( b \), without alterations in the base.

For instance, if we consider the function \( y = 2(1.5)^t \), we can easily see your \( b \) is 1.5, confirming exponential growth. Numerous real-world scenarios follow such a pattern, making this an important concept to comprehend.

Exponential Decay

Exponential decay describes a process where a quantity decreases rapidly in proportion to its current value over time. In the general form of an exponential function \( y = a (b)^t \), for decay, the base \( b \) is between 0 and 1. This means every increase in the variable \( t \) results in a consistent reduction of \( y \), leading to a rapid decline initially that slows down over time.
This type of decay happens often in nature, such as with radioactive materials and cooling processes.
### Recognizing Exponential Decay:

• The base \( b \) needs to be between 0 and 1.
• The exponent \( t \) is directly applied to the base \( b \).

Consider, for example, the function \( y = 500(0.8)^t \). Here, the base is 0.8, confirming exponential decay. Understanding these principles can help you predict how quickly systems lose value or quantity over time.

Equation Analysis

To determine if an equation represents exponential growth or decay, it's crucial to analyze the structure of the equation. The standard form is \( y = a (b)^t \) with \( t \) as the exponent. In the given equation \( y = 300(1-t)^5 \), notice that the variable \( t \) is inside the base, not as an exponent. This deviates from the norm as \( t \) should directly influence the power for exponential forms.
### Key Features in Analyzing:

• Check if the base \( b \) is raised to the exponent \( t \) - an essential feature of exponential functions.
• Ensure that \( b \) remains constant and doesn't change with \( t \).

This structural irregularity means the given equation doesn't fit into either exponential growth or decay. For a clearer categorical understanding, compare equations to the standard form, ensuring \( t \) directly impacts the exponentiation of \( b \). Recognizing these patterns will help in identifying and analyzing these forms accurately.