Problem 1 - Precalculus: A Concise Course

Question

An exponential growth model has the form ________ and an exponential decay model has the form ________.

Step-by-Step Solution

Answer

The exponential growth model is given by the formula \( y = a * b^{(t)} \) and the exponential decay model is given by the formula \( y = a * (1-b)^{t} \).

Step 1: Exponential Growth Function

An exponential growth model is represented by the formula: \( y = a * b^{(t)} \), where: \n- \( a \) is the initial value \n- \( b \) is the growth factor \n- \( t \) is the time

Step 2: Exponential Decay Function

An exponential decay model is represented by the formula: \( y = a * (1-b)^{t} \), where:\n- \( a \) is the initial value \n- \( b \) is the decay factor\n- \( t \) is the time. Here the factor \( (1-b) \) is less than 1, which ensures the value of \( y \) decreases over time, hence representing decay.

Key Concepts

Exponential growth

Exponential growth occurs when the quantity increases by a fixed percentage over a consistent time period. Imagine a scenario where you place money in a savings account with a certain yearly interest rate. Each year, the money grows by that interest rate, and this is an example of exponential growth.
In mathematical terms, exponential growth is given by the function \( y = a \times b^{t} \).
Here:

• \( a \) is the starting quantity or initial value.
• \( b \) is the growth factor, which is greater than 1.
• \( t \) is the time period.

The key characteristic of exponential growth is how growth accelerates as time goes on. This type of growth is unsustainable in the long-term as resources or space may run out, like a bacteria culture multiplying rapidly until the nutrients are depleted.

Exponential decay

Exponential decay is the process where a quantity decreases by a constant percentage over time. Think of it like a melting ice cube. The ice quickly shrinks in the beginning but the pace slows down as it gets smaller.
In mathematical terms, exponential decay is given by the function \( y = a \times (1-b)^{t} \).
Here:

• \( a \) is the original value or quantity.
• \( b \) is the decay factor, which is a positive number, less than 1, ensuring the rate of reduction.
• \( t \) is the time period.

With decay, each successive time period results in a smaller percentage of the original value decreasing. This is why exponential decay often models processes like radioactive decay or depreciation of assets, where the reduction becomes less noticeable over time.

Growth factor

The growth factor is a crucial part of understanding how quickly something grows exponentially. It is represented by the symbol \( b \) in an exponential function. For exponential growth, this factor \( b \) is always greater than 1. This ensures that the quantity grows with each passing time period.
The growth factor directly correlates to the percentage increase per time unit. For instance, a growth factor of 1.03 represents a 3% increase each period. The relationship is:\[b = 1 + \text{growth rate (as a decimal)}\]This compact number is paramount because even small seeming growth factors can lead to large increases over long periods due to the cumulative effect of compounding. That's why understanding compound interest, population growth, or viral spread often requires comprehension of growth factors.

Decay factor

The decay factor is a key component in understanding exponential decay. Notated as \( b \) in the decay formula, it expresses how quickly something decreases over time. For exponential decay, this factor is between 0 and 1, as it denotes a percentage decrease.Understanding this factor means realizing that your quantity grows smaller by a consistent proportion each period. With a decay factor \( b \), you use the relationship:\[b = 1 - \text{decay rate (as a decimal)}\]For example, a decay factor of 0.97 represents a 3% decrease.By integrating this concept into scenarios like depreciation of values in finance, dying populations in biology, or decreasing charges in physics, you gain insight into how prolonged effects translate into gradual reduction.