Problem 13 - Algebra 1: Concepts and Skills

Question

Identify the initial amount and the decay factor in the exponential function. $$ y=18(0.11)^{t} $$

Step-by-Step Solution

Answer

The initial amount 'a' is 18, and the decay factor 'b' is 0.11.

Step 1: Understand the exponential decay equation

The basic form of an exponential decay function is \(y = a(b)^{t}\). Here, 'a' is the initial amount (which can also be called the starting value or the y-intercept). 'b' is the base or the decay factor, and 't' is the variable (like time).

Step 2: Identify the initial amount

In the given function \(y = 18(0.11)^{t}\), the coefficient of the exponential term (the number multiplied by the base raised to the power of t) is 18. Hence, the initial amount, 'a', is 18.

Step 3: Identify the decay factor

The base of the exponent in the given function is 0.11, that's the number b. Since it is less than 1, it indicates that the function represents a decay, or decrease, over time. Hence, the decay factor, 'b', is 0.11.

Key Concepts

Initial Amount

In an exponential decay function, one of the key components is the initial amount. It represents the starting point or original value before any decay occurs. Think of it like the full value of a battery when it is fully charged. In mathematical terms, it's symbolized by 'a' in the equation. This is crucial because it sets the baseline from which the decay factor will operate.

In the context of the given exponential function \(y = 18(0.11)^{t}\), the initial amount is 18. This means that at the start, before any time passes or decay happens, the quantity begins at 18 units. It's like saying at time zero, your starting position is 18. This value provides a clear reference point, helping to measure how much is left as time progresses.

Decay Factor

The decay factor in an exponential decay function is the value that determines how quickly the initial amount decreases as time passes. Whenever this factor is less than one, it signals that the quantity is decreasing over time. This concept is reflected in the equation as the base of the exponential term, often represented by 'b'.

In the function \(y = 18(0.11)^{t}\), the decay factor is 0.11. Here's how it works:

• The decay factor of 0.11 means that with each unit of time that passes (for instance, each hour or day), the initial amount multiplies by 0.11.
• This consistent multiplication causes the quantity to shrink, moving closer to zero with every time interval.

Simply put, the smaller this factor, the faster the decay. The nature of an exponential function means that the reduction in value doesn't happen linearly but rather more dramatically as time continues, due to the multiplying effect.

Exponential Function

An exponential function is a mathematical expression that models situations where quantities grow or decay at coconstant relative rates. These functions include a constant raised to the power of a variable, usually to represent time. They are powerful tools for representing real-world phenomena, such as population growth, radioactive decay, or even financial investments. The form of an exponential function is typically given as \(y = a(b)^{t}\).

In this form:

• '\(a\)' represents the initial amount.
• '\(b\)' is the base, also known as the growth or decay factor.
• '\(t\)' signifies the time or the independent variable.

In our specific function \(y = 18(0.11)^{t}\), since the base is less than 1, it indicates an exponential decay. This means that as the variable \(t\) increases, the value of \(y\) decreases, illustrating how the initial amount diminishes over time. Understanding this behavior is essential for interpreting data in fields ranging from science to economics, wherever there's a pattern of decrease over time.