\( \left. \begin{array} { l } { y = a ( 1 - r ) ^ { t } } \\ { y = 100,000 ( 1 - 0.02 ) ^ { t } } \\ { y = 100,000 ( 0.98 ) ^ { t } - 4 } \end{array} \right. \)

It appears you're working with a series of equations that model exponential decay. Let's break down each equation to understand their relationships and implications. ### 1. General Exponential Decay Formula \[ y = a \cdot (1 - r)^t \] - **\( y \)**: The quantity at time \( t \). - **\( a \)**: The initial amount at time \( t = 0 \). - **\( r \)**: The decay rate per time period. - **\( t \)**: The number of time periods. **Explanation:** This equation models how a quantity decreases over time at a constant percentage rate \( r \). Each time period, the quantity retains \( (1 - r) \) of its value from the previous period. --- ### 2. Specific Instance of the Decay Formula \[ y = 100,\!000 \cdot (1 - 0.02)^t \] Simplified: \[ y = 100,\!000 \cdot 0.98^t \] - **\( a = 100,\!000 \)**: The initial amount is 100,000. - **\( r = 0.02 \)**: The decay rate is 2% per time period. **Explanation:** Here, the equation specifies that the initial amount is 100,000, and it decays by 2% each time period. After each period \( t \), the quantity becomes 98% of its value from the previous period. --- ### 3. Modified Decay Equation \[ y = 100,\!000 \cdot 0.98^t - 4 \] - **Additional Term \( -4 \)**: A constant subtraction of 4 units is introduced. **Explanation:** This equation introduces a **fixed decrease** of 4 units **in addition** to the exponential decay. This means that for every time period \( t \), not only does the quantity decay by 2%, but it also decreases by an additional 4 units. **Possible Interpretations:** 1. **Combined Decay and Fixed Reduction:** If there's a scenario where the quantity is subject to both a percentage-based decay and a constant reduction (e.g., depreciation plus maintenance costs), this equation would make sense. 2. **Typographical Error:** If the \( -4 \) was not intended, it might be a mistake. Ensure that this term is necessary for your model. 3. **Linear vs. Exponential Terms:** Combining exponential and linear terms can model more complex behaviors but also complicates the interpretation and analysis. --- ### Considerations and Next Steps 1. **Purpose of the Model:** - **Pure Exponential Decay:** If you're modeling a situation where only the percentage-based decay is relevant (like radioactive decay), the second equation suffices. - **Combined Effects:** If there's an additional constant factor affecting the quantity, the third equation might be appropriate. 2. **Analyzing the Impact of the \( -4 \) Term:** - **Long-Term Behavior:** Over time, the constant subtraction can have a more significant impact than the exponential term, potentially driving \( y \) negative if not bounded. - **Practical Constraints:** Ensure that the model remains realistic within the context it's applied. For example, if \( y \) represents a population or a financial quantity, negative values might not make sense. 3. **Graphical Representation:** Plotting both equations can help visualize the differences: - **Exponential Decay Only:** A smooth curve approaching zero. - **Exponential Decay with Constant Subtraction:** A curve that approaches \( -4 \) as \( t \) increases. 4. **Solving for Variables:** If you need to solve for \( t \) given a specific \( y \), or vice versa, you can manipulate the equations accordingly: - **For Eq. 2:** \[ t = \frac{\ln(y/a)}{\ln(1 - r)} = \frac{\ln(y/100,\!000)}{\ln(0.98)} \] - **For Eq. 3:** Solving for \( t \) becomes more complex due to the constant term and may not have an analytical solution, requiring numerical methods. 5. **Clarify Your Objective:** If you have a specific goal (e.g., determining when \( y \) falls below a threshold, comparing models, etc.), providing that context can lead to more targeted assistance. --- **Feel free to provide additional context or specify how you'd like to proceed with these equations. Whether it's solving for a variable, interpreting results, or extending the model, I'm here to help!**