2010 there were 10 frogs, in 2021 there is 20 frogs. What is the average growth rate per year? a) What is the predicted amount of frogs in 2030? 

We can solve this problem by using the formula for exponential growth, which is:

\[
P(t) = P_0 \cdot (1 + r)^t
\]

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population.
- \( r \) is the growth rate.
- \( t \) is the time in years.

### Step 1: Calculate the growth rate
Given:
- Initial population in 2010: \( P_0 = 10 \) frogs.
- Population in 2021: \( P(t) = 20 \) frogs.
- Time difference \( t = 2021 - 2010 = 11 \) years.

Using the formula for exponential growth:

\[
20 = 10 \cdot (1 + r)^{11}
\]

Divide both sides by 10:

\[
2 = (1 + r)^{11}
\]

Take the 11th root of both sides:

\[
1 + r = 2^{\frac{1}{11}}
\]

Now calculate the growth rate \( r \):

\[
r = 2^{\frac{1}{11}} - 1
\]

### Step 2: Predict the frog population in 2030
Once we know the growth rate \( r \), we can use the same exponential growth formula to predict the population in 2030.

We have:
- \( P_0 = 10 \) (initial population in 2010),
- \( r \) (from the previous step),
- Time difference \( t = 2030 - 2010 = 20 \) years.

We will plug these values into the exponential growth formula to find the predicted population \( P(20) \).

Let me compute the results.### Results:

1. The average annual growth rate is approximately \( r = 0.065 \) or 6.5%.
2. The predicted frog population in 2030 is about 35 frogs.

Would you like more details or have any further questions?

Here are five related questions to expand this topic:
1. How does the calculation change if the population doubles in 8 years instead of 11?
2. What if the frog population grows according to a linear model instead of exponential?
3. How would the predictions change if the environment limits growth after a certain population size?
4. Can this model apply to other species with different lifespans or reproduction rates?
5. How accurate is this model in the long term for small animal populations?

**Tip:** Exponential growth models assume continuous, unlimited growth, which might not always reflect real-world conditions.

2010 there were 10 frogs, in 2021 there is 20 frogs. What is the average growth rate per year? a) What is the predicted amount of frogs in 2030?

Calculate the average annual growth rate of a frog population that grows from 10 frogs in 2010 to 20 frogs in 2021 using exponential growth formulas. Predict the population size in 2030. Suitable for students in Grades 10-12.

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