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

Question

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

UpStudy AI · Accepted Answer

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!**
The equations model exponential decay with an initial amount of 100,000, decaying at a 2% rate per period. The third equation adds a constant decrease of 4 units each period.

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

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