The exponential growth can be mathematically expressed as

Options

A. `L_(t) = L_(0) +rt`
B. `W_(1) = W_(0) +e^(rt)`
C. `W_(1) = W_(0) e^(rt)`
D. `L_(t) = L_(0) - rt`

Answer

To express exponential growth mathematically, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Exponential Growth**: Exponential growth occurs when the growth rate of a value is proportional to its current value. This means that as the value increases, the rate of growth also increases.

2. **Mathematical Expression**: The mathematical expression for exponential growth can be represented as:
\[ W_1 = W_0 e^{(r \cdot t)} \]
where:
- \( W_1 \) = final size (could refer to weight, height, population number, etc.)
- \( W_0 \) = initial size (the size at the beginning of the observation period)
- \( r \) = growth rate (the rate at which the growth occurs)
- \( t \) = time (the duration over which the growth is observed)
- \( e \) = base of the natural logarithm (approximately equal to 2.71828)

3. **Components of the Equation**:
- **Final Size (\( W_1 \))**: This represents the outcome after a certain period of growth.
- **Initial Size (\( W_0 \))**: This is the starting point before any growth has occurred.
- **Growth Rate (\( r \))**: This indicates how quickly the growth is happening relative to the current size.
- **Time (\( t \))**: This is the duration for which the growth is being measured.

4. **Understanding the Growth Rate**: The growth rate \( r \) is crucial as it determines how effectively the plants can produce biomass or increase in size over time. It can be viewed as an efficiency index for the growth of the plant.

5. **Conclusion**: Therefore, the correct mathematical expression for exponential growth is given by option number 3:
\[ W_1 = W_0 e^{(r \cdot t)} \]