3. The graph shows the amount of a chemical in a water sample it is decreasing exponentially. Find the coordinates of the points labeled $$ A, B $$, and $$ C $$. Explain your reasoning

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To determine the coordinates of points $$ A $$, $$ B $$, and $$ C $$ on an exponential decay graph representing the amount of a chemical in a water sample over time, we'll first understand the general behavior of exponential decay and then apply this understanding to identify the key points.

### Understanding Exponential Decay

Exponential decay can be modeled by the equation:

$$
y(t) = y_0 \cdot e^{-kt}
$$

Where:
- $$ y(t) $$ is the amount of the chemical at time $$ t $$.
- $$ y_0 $$ is the initial amount of the chemical at $$ t = 0 $$.
- $$ k $$ is the decay constant (a positive number).
- $$ e $$ is Euler's number (approximately 2.71828).

Key characteristics of the graph:
- **Initial Value**: At $$ t = 0 $$, the amount is $$ y_0 $$.
- **Asymptote**: The graph approaches but never touches the horizontal axis ($$ y = 0 $$).
- **Half-Life**: The time it takes for the amount to reduce to half its initial value.

### Identifying Points $$ A $$, $$ B $$, and $$ C $$

Assuming the points are labeled at significant moments in the decay process, here's how you might identify their coordinates:

1. **Point $$ A $$: Initial Amount**
   - **Location**: At time $$ t = 0 $$.
   - **Coordinates**: $$ (0, y_0) $$.
   - **Reasoning**: This represents the starting quantity of the chemical before any decay has occurred.

2. **Point $$ B $$: Amount After One Half-Life**
   - **Location**: At time $$ t = t_{1/2} $$ (one half-life).
   - **Coordinates**: $$ (t_{1/2}, \frac{y_0}{2}) $$.
   - **Reasoning**: After one half-life, the amount of the chemical decreases to half its initial value.

3. **Point $$ C $$: Amount After Two Half-Lives**
   - **Location**: At time $$ t = 2t_{1/2} $$ (two half-lives).
   - **Coordinates**: $$ (2t_{1/2}, \frac{y_0}{4}) $$.
   - **Reasoning**: After two half-lives, the amount decreases to one-quarter of the initial value.

### Example

Let's assume the initial amount $$ y_0 $$ is **100 grams**, and the half-life $$ t_{1/2} $$ is **5 hours**.

- **Point $$ A $$**: $$ (0, 100) $$
- **Point $$ B $$**: $$ (5, 50) $$
- **Point $$ C $$**: $$ (10, 25) $$

### Visualization

Here's a rough sketch of how these points appear on the graph:

```
Amount
 |               
100 |A
 |                
 |                
50  |        B
 |                 
 |                
25  |                C
 |____________________________ Time
    0    5      10      15 ...
```

### Summary

- **Point $$ A $$** is always at $$ (0, y_0) $$, representing the initial quantity.
- **Point $$ B $$** is at $$ (t_{1/2}, y_0/2) $$, showing the amount after one half-life.
- **Point $$ C $$** is at $$ (2t_{1/2}, y_0/4) $$, indicating the amount after two half-lives.

By identifying key moments in the decay process—such as the initial time, one half-life, and subsequent half-lives—you can accurately determine the coordinates of points labeled on an exponential decay graph.
Points $$ A $$, $$ B $$, and $$ C $$ are located at $$ (0, y_0) $$, $$ (t_{1/2}, \frac{y_0}{2}) $$, and $$ (2t_{1/2}, \frac{y_0}{4}) $$ respectively, where $$ y_0 $$ is the initial amount and $$ t_{1/2} $$ is the half-life of the chemical.

The graph shows the amount of a chemical in a water sample it is decreasing
exponentially.
Find the coordinates of the points labeled , and . Explain your reasoning

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Understanding Exponential Decay

Identifying Points , , and

Example

Visualization

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The graph shows the amount of a chemical in a water sample it is decreasing exponentially. Find the coordinates of the points labeled , and . Explain your reasoning