Problem 14 - College Algebra with Corequisite Support

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 Exponential Model

Recognize the standard form of an exponential function, which is typically in the form of either \[ y = a(1 + r)^t \] for growth, or \[ y = a(1 - r)^t \] for decay, where \( a \) is the initial amount, \( r \) is the rate of growth or decay, and \( t \) is time.

Step 2: Analyze the Equation

Compare the given equation \[ y = 300(1-t)^5 \] to the standard form of exponential functions. Observe that in the given equation, \( t \) is part of the base of the exponent, making this equation non-standard.

Step 3: Determine if It Is Growth, Decay, or Neither

The term \( 1-t \) does not fit the standard form for exponential growth or decay due to the negative influence of \( t \) within the base of the exponent. This makes the function behave in a manner that is neither consistent with exponential growth \( (1+r) \) nor exponential decay \( (1-r) \) because the base can vary in an unpredictable way.

Key Concepts

Exponential Growth

Exponential growth occurs when the increase of a quantity is proportional to its current value. In simpler terms, exponential growth happens when something keeps growing at a constant percentage rate over time. A familiar analogy is a bank account that earns interest over and over. Each year, interest is calculated on the current value, compounding the growth.
For exponential growth, the general formula is \( y = a(1 + r)^t \). In this formula:

• \( y \) is the final amount after time \( t \).
• \( a \) is the initial amount or starting value.
• \( r \) is the growth rate, represented as a decimal.
• \( t \) stands for time, which can be in years, months, etc.

As time progresses, \( y \) grows larger and larger if \( r > 0 \). The higher the rate, the faster the growth. Exponential growth is not linear; it speeds up over time, like a snowball rolling down a hill, gathering more snow and getting bigger as it goes.

Exponential Decay

Exponential decay, on the other hand, describes a process where a quantity decreases at a constant percentage rate over time. This is often observed in radioactive decay or in the depreciation of assets like cars.
The standard formula used for exponential decay is \( y = a(1 - r)^t \). Here:

• \( y \) represents the amount remaining after time \( t \).
• \( a \) is the initial amount or starting value.
• \( r \) is the decay rate, and it must be between 0 and 1.
• \( t \) is the time that has passed.

With each time period, the quantity becomes smaller if \( 0 < r \leq 1 \). As time goes on, the value decreases at a progressively slower rate, which makes exponential decay a bit like watching a melting ice cube - it shrinks steadily but not in a straight line.

Standard Form of Exponential Functions

The standard form of exponential functions is essential for understanding whether an equation exhibits exponential growth or decay. As mentioned in earlier sections, an exponential function typically appears in one of these forms:

• Exponential Growth: \( y = a(1 + r)^t \)
• Exponential Decay: \( y = a(1 - r)^t \)

The distinction between growth and decay hinges on the term within the parentheses:

• If it is \((1 + r)\), the function depicts growth.
• If it is \((1 - r)\), it signals decay.

The equation from the exercise, \(y = 300(1-t)^5\), does not fit neatly into this standard form since the variable \(t\) directly alters the base value \((1-t)\) unpredictably, instead of acting as an exponent on a fixed base as seen in standard forms. Consequently, such a function does not exhibit constant proportional growth or decay, making it neither exponential growth nor decay.