Question
Which equations and graphs represent exponential decay?
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Answers
GPT-4o mini
Exponential decay occurs when a quantity decreases at a rate proportional to its current value. This can be mathematically represented by the equation:
\[
y(t) = y_0 e^{-kt}
\]
Where:
- \( y(t) \) is the quantity at time \( t \)
- \( y_0 \) is the initial quantity (when \( t = 0 \))
- \( k \) is a positive constant that determines the rate of decay
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828)
### Examples of Exponential Decay Equations:
1. **Basic Form:**
\[
y = 5 e^{-0.1t}
\]
This represents a quantity starting at 5 and decaying at a rate determined by the constant \( 0.1 \).
2. **Half-Life Form:**
\[
N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
\]
This form is often used in contexts like radioactive decay, where \( T_{1/2} \) is the half-life of the substance.
### Graph Characteristics:
- The graph of an exponential decay function will show a curve that starts at the value of \( y_0 \) when \( t = 0 \) and approaches zero as \( t \) increases.
- The graph will never actually reach zero; this is known as an asymptotic behavior.
### Key Features of the Graph:
- **Y-Intercept:** The y-intercept of the graph will be at \( (0, y_0) \).
- **Decay Rate:** A larger value of \( k \)/smaller \( T_{1/2} \) leads to a steeper slope and faster decay.
- **Asymptote:** The x-axis (y = 0) serves as a horizontal asymptote.
### Graph Example:
Graphing the function \( y(t) = 5 e^{-0.1t} \) will show a decreasing curve starting at \( y = 5 \) at \( t = 0 \), gradually approaching the x-axis but never actually reaching it.
In summary, exponential decay is represented by equations of the form \( y(t) = y_0 e^{-kt} \), and the graphs will depict a continuous decrease towards zero over time, characterized by a smooth, downward-sloping curve.
\[
y(t) = y_0 e^{-kt}
\]
Where:
- \( y(t) \) is the quantity at time \( t \)
- \( y_0 \) is the initial quantity (when \( t = 0 \))
- \( k \) is a positive constant that determines the rate of decay
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828)
### Examples of Exponential Decay Equations:
1. **Basic Form:**
\[
y = 5 e^{-0.1t}
\]
This represents a quantity starting at 5 and decaying at a rate determined by the constant \( 0.1 \).
2. **Half-Life Form:**
\[
N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
\]
This form is often used in contexts like radioactive decay, where \( T_{1/2} \) is the half-life of the substance.
### Graph Characteristics:
- The graph of an exponential decay function will show a curve that starts at the value of \( y_0 \) when \( t = 0 \) and approaches zero as \( t \) increases.
- The graph will never actually reach zero; this is known as an asymptotic behavior.
### Key Features of the Graph:
- **Y-Intercept:** The y-intercept of the graph will be at \( (0, y_0) \).
- **Decay Rate:** A larger value of \( k \)/smaller \( T_{1/2} \) leads to a steeper slope and faster decay.
- **Asymptote:** The x-axis (y = 0) serves as a horizontal asymptote.
### Graph Example:
Graphing the function \( y(t) = 5 e^{-0.1t} \) will show a decreasing curve starting at \( y = 5 \) at \( t = 0 \), gradually approaching the x-axis but never actually reaching it.
In summary, exponential decay is represented by equations of the form \( y(t) = y_0 e^{-kt} \), and the graphs will depict a continuous decrease towards zero over time, characterized by a smooth, downward-sloping curve.