Question

. Read each of the following descriptions, and identify the quantities that are
undergoing exponential growth.
• the amount of radioactive material in a 30-gram sample that decays
by 25% each day
• the value of a comic book that initially cost $3.99 increases by 5%
each year
• the value of a stock initially priced at $2.75 increases by 2.5% each
month
• the population of a city of 45,000 decreases by 3% each year
B. For each description in part A that represents exponential growth, write an
exponential equation to model the relationship between the two quantities.
C. For each equation in part B, explain the meaning of x, y, a, and b in the
context of the situation described.
D. For each equation in part B, create a table of values, and use the table of
values to graph the relationship between the two quantities.

Answer 1

Let’s analyze each of the descriptions to identify the instances of exponential growth.

### Part A: Identifying Exponential Growth

1. **Radioactive Material (30-gram sample, decays by 25% each day)**
- **Not exponential growth**: It's exponential decay, as the amount decreases over time.

2. **Comic Book Value (initially $3.99, increases by 5% each year)**
- **Exponential growth**: The value increases over time.

3. **Stock Value (initially priced at $2.75, increases by 2.5% each month)**
- **Exponential growth**: The value increases over time.

4. **Population of a City (45,000, decreases by 3% each year)**
- **Not exponential growth**: It's exponential decay, as the population decreases over time.

**Exponential Growth Quantities Identified:** Comic book value and stock value.

### Part B: Exponential Equations

For the quantities identified as undergoing exponential growth, we can write the equations as follows:

1. **Comic Book Value**
- Let \( y \) represent the value of the comic book after \( x \) years.
- Initial value \( a = 3.99 \).
- Growth rate \( b = 1 + 0.05 = 1.05 \).
**Exponential Equation:**
\( y = 3.99(1.05)^x \)

2. **Stock Value**
- Let \( y \) represent the value of the stock after \( x \) months.
- Initial value \( a = 2.75 \).
- Growth rate \( b = 1 + 0.025 = 1.025 \).
**Exponential Equation:**
\( y = 2.75(1.025)^x \)

### Part C: Explanation of Variables

1. **Comic Book Value Equation: \( y = 3.99(1.05)^x \)**
- \( x \): The number of years since the initial purchase of the comic book.
- \( y \): The value of the comic book after \( x \) years.
- \( a = 3.99 \): The initial value of the comic book.
- \( b = 1.05 \): The growth factor, indicating that the value increases by 5% each year.

2. **Stock Value Equation: \( y = 2.75(1.025)^x \)**
- \( x \): The number of months since the stock was purchased.
- \( y \): The value of the stock after \( x \) months.
- \( a = 2.75 \): The initial value of the stock.
- \( b = 1.025 \): The growth factor, indicating that the value increases by 2.5% each month.

### Part D: Table of Values

**Table for Comic Book Value (\( y = 3.99(1.05)^x \)):**

| \( x \) (Years) | \( y \) (Value) |
|------------------|----------------------|
| 0 | \( 3.99(1.05)^0 = 3.99 \) |
| 1 | \( 3.99(1.05)^1 \approx 4.19 \) |
| 2 | \( 3.99(1.05)^2 \approx 4.40 \) |
| 3 | \( 3.99(1.05)^3 \approx 4.62 \) |
| 4 | \( 3.99(1.05)^4 \approx 4.85 \) |
| 5 | \( 3.99(1.05)^5 \approx 5.09 \) |

**Table for Stock Value (\( y = 2.75(1.025)^x \)):**

| \( x \) (Months) | \( y \) (Value) |
|-------------------|---------------------|
| 0 | \( 2.75(1.025)^0 = 2.75 \) |
| 1 | \( 2.75(1.025)^1 \approx 2.82 \) |
| 2 | \( 2.75(1.025)^2 \approx 2.89 \) |
| 3 | \( 2.75(1.025)^3 \approx 2.96 \) |
| 4 | \( 2.75(1.025)^4 \approx 3.03 \) |
| 5 | \( 2.75(1.025)^5 \approx 3.11 \) |

### Graphing the Relationships

Now, using the tables above, you can plot the values on a graph for both the comic book value over years and the stock value over months:

- **X-axis** will represent time (years for comic book and months for stock).
- **Y-axis** will represent the value.

You can use any graphing tool to visualize these relationships, confirming the exponential growth nature shown by the increasing slope of the curves in both cases.