An **exponential growth pattern** occurs when the growth rate of a quantity is proportional to its current value, meaning that the quantity grows by a fixed percentage (or factor) over equal time intervals. This pattern can be modeled mathematically by the exponential function:

\[
P(t) = P_0 \cdot e^{rt}
\]

Where:
- \(P(t)\) is the amount of the quantity at time \(t\),
- \(P_0\) is the initial amount (at time \(t = 0\)),
- \(r\) is the growth rate (expressed as a decimal),
- \(t\) is the time that has passed,
- \(e\) is the base of the natural logarithm (approximately 2.71828).

### Key Features of Exponential Growth:
1. **Continuous Growth**: The quantity increases continuously at a rate proportional to its size.
2. **Doubling Time**: The time it takes for the quantity to double can be determined using the rule of 70. If the growth rate is \(r\) (as a percentage), the doubling time \(T_d\) is approximately:
   
   \[
   T_d = \frac{70}{r}
   \]

3. **Accelerating Growth**: Exponential growth accelerates as the quantity increases, leading to very rapid increases after a certain point.
4. **Growth Factor**: In some cases, the formula can be written as \( P(t) = P_0 \cdot (1 + r)^t \) when considering discrete growth intervals.

### Example:
Suppose a population of bacteria doubles every hour. If the initial population is 100 bacteria, the growth rate \(r\) would be 100% (or 1.0 in decimal form), and the function describing the population over time would be:

\[
P(t) = 100 \cdot e^{1t}
\]

After 3 hours, the population would be:

\[
P(3) = 100 \cdot e^3 \approx 100 \cdot 20.0855 = 2008.55 \text{ bacteria}
\]

This shows how the population grows exponentially over time.

### Do you want details or have any questions?

Here are 5 related questions to explore further:
1. What is the difference between exponential growth and linear growth?
2. How can exponential decay be modeled similarly to exponential growth?
3. Can exponential growth continue indefinitely in real-world scenarios?
4. How do you calculate the time needed to reach a specific value in exponential growth?
5. How does compound interest work as a real-life example of exponential growth?

**Tip**: Always check the base of your exponential function. If the base is **e**, it's continuous growth, and if the base is a fixed number like 2, it might indicate discrete intervals.

Explore the concept of exponential growth with detailed examples and key formulas. Learn how to use the exponential growth equation P(t) = P_0 * e^{rt} and calculate doubling time using the Rule of 70. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation