The exponential growth can be expressed as

Options

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

Answer

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

1. **Identify the Variables**: In the context of exponential growth, we have:
- \( W_t \): the final size or population at time \( t \)
- \( W_0 \): the initial size or population at time \( t = 0 \)
- \( r \): the growth rate (expressed as a decimal)
- \( t \): the time duration
- \( e \): the base of the natural logarithm, approximately equal to 2.71828

2. **Write the Exponential Growth Formula**: The general formula for exponential growth can be expressed as:
\[ W_t = W_0 \cdot e^{rt} \]
Here, \( W_t \) is determined by multiplying the initial size \( W_0 \) by \( e \) raised to the power of the product of the growth rate \( r \) and time \( t \).

3. **Interpret the Components**:
- \( e^{rt} \) represents the growth factor over time \( t \) at a constant growth rate \( r \).
- As time increases, the term \( e^{rt} \) increases exponentially, leading to rapid growth in the size or population.

4. **Conclusion**: Therefore, the expression for exponential growth is:
\[ W_t = W_0 \cdot e^{rt} \]