Exponential growth can be expressed as

Options

A. `W_1=W_0e^(rt)`
B. `W_0=W_1e^(rt)`
C. `W_1=W_0e^(Deltart)`
D. `W_0=W_1e^(Deltart)`

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. In biological terms, this means that as a plant grows, the rate at which it grows increases as its size increases.

2. **Identifying the Variables**:
- Let \( W_t \) be the final weight of the plant after a certain time \( t \).
- Let \( W_0 \) be the initial weight of the plant.
- Let \( r \) be the growth rate (also known as the efficiency index).
- Let \( t \) be the time duration for which we are observing the growth.
- The base of the natural logarithm is denoted by \( e \).

3. **Formulating the Exponential Growth Equation**:
- The general formula for exponential growth can be expressed as:
\[ W_t = W_0 \cdot e^{rt} \]
- Here, \( W_t \) is the weight at time \( t \), \( W_0 \) is the initial weight, \( r \) is the growth rate, and \( t \) is the time.

4. **Interpreting the Equation**:
- In this equation, as time \( t \) increases, the weight \( W_t \) increases exponentially if \( r \) is positive.
- This reflects the characteristic of exponential growth where the growth accelerates over time.

5. **Conclusion**:
- Therefore, the expression for exponential growth can be succinctly stated as:
\[ W_t = W_0 \cdot e^{rt} \]