The number of bats in a colony is growing exponentially. After  
1
  year, there were  
70
  bats. After  
4
  years, there were  
560
  bats.

If the colony continues to grow at the same rate, how many bats are expected to be in the colony after  
7
  years? Do not include units in your answer.

To solve this problem, we will model the population of bats using the exponential growth formula:

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

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

We are given:
- After 1 year, \(P(1) = 70\),
- After 4 years, \(P(4) = 560\).

### Step 1: Set up two equations based on the data.

For \(t = 1\):
\[
P(1) = P_0 \cdot e^{k \cdot 1} = 70 \quad \Rightarrow \quad P_0 \cdot e^k = 70
\]

For \(t = 4\):
\[
P(4) = P_0 \cdot e^{k \cdot 4} = 560 \quad \Rightarrow \quad P_0 \cdot e^{4k} = 560
\]

### Step 2: Solve the system of equations.

To eliminate \(P_0\), divide the second equation by the first:
\[
\frac{P_0 \cdot e^{4k}}{P_0 \cdot e^k} = \frac{560}{70}
\]

This simplifies to:
\[
e^{3k} = 8
\]

Take the natural logarithm (ln) of both sides:
\[
3k = \ln(8)
\]
\[
k = \frac{\ln(8)}{3} \approx \frac{2.0794}{3} \approx 0.6931
\]

### Step 3: Find \(P_0\).

Using the first equation \(P_0 \cdot e^k = 70\), substitute \(k \approx 0.6931\):
\[
P_0 \cdot e^{0.6931} = 70 \quad \Rightarrow \quad P_0 \cdot 2 = 70 \quad \Rightarrow \quad P_0 = \frac{70}{2} = 35
\]

### Step 4: Calculate the population after 7 years.

Now that we know \(P_0 = 35\) and \(k \approx 0.6931\), we can find the population after 7 years using the formula \(P(7) = P_0 \cdot e^{k \cdot 7}\):

\[
P(7) = 35 \cdot e^{0.6931 \cdot 7} = 35 \cdot e^{4.8517} \approx 35 \cdot 127.017 \approx 4445.6
\]

So, the population after 7 years is approximately **4446** bats.

Would you like further clarification or additional details?

### Related Questions:
1. How does exponential growth differ from linear growth?
2. Can exponential growth continue indefinitely in real-world scenarios?
3. How does the growth rate affect the speed of population increase?
4. How would the model change if we had a carrying capacity in the environment?
5. What is the doubling time for this colony of bats?

### Tip:
Always double-check calculations when dealing with exponentials, especially when approximating values.

The number of bats in a colony is growing exponentially. After 1 year, there were 70 bats. After 4 years, there were 560 bats. If the colony continues to grow at the same rate, how many bats are expected to be in the colony after 7 years?

Learn how to model the exponential growth of a bat colony using the exponential growth formula. Starting with 70 bats after 1 year and 560 bats after 4 years, calculate the colony's population after 7 years with a step-by-step guide. Suitable for high school students.

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