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The image contains two problems from a math assignment.

**Problem 6**:
The population of fish is given by the equation:
\[ P(t) = 3.2 \times 10^4 \cdot 1.32^t \]
The question asks: when will the population reach 1 million and 1 billion?

**Problem 7**:
The equation for population growth is given by:
\[ P(t) = 1200 \cdot 1.32^t \]
The task is to find the doubling time of the population.

Let me start by solving **Problem 6**.

We need to find when the population \( P(t) \) reaches 1 million and 1 billion. To do this, we will solve the equation for \( t \) when \( P(t) = 1,000,000 \) and \( P(t) = 1,000,000,000 \).

The equation is:
\[
P(t) = 3.2 \times 10^4 \cdot 1.32^t
\]
For \( P(t) = 1,000,000 \):
\[
1,000,000 = 3.2 \times 10^4 \cdot 1.32^t
\]
Solving for \( t \), we get:
\[
\frac{1,000,000}{3.2 \times 10^4} = 1.32^t
\]
\[
\frac{1,000,000}{32,000} = 1.32^t
\]
\[
31.25 = 1.32^t
\]
Now take the natural logarithm of both sides:
\[
\ln(31.25) = t \ln(1.32)
\]
\[
t = \frac{\ln(31.25)}{\ln(1.32)}
\]

I'll calculate this value.For **Problem 6**:

- The population will reach **1 million** after approximately **12.40 years**.
- The population will reach **1 billion** after approximately **37.28 years**.

Next, let's solve **Problem 7** to find the doubling time for the population. The equation is:
\[
P(t) = 1200 \cdot 1.32^t
\]
We need to find \( t \) such that the population doubles, i.e., when \( P(t) = 2 \times 1200 = 2400 \).

Let me calculate this next.For **Problem 7**, the population will double approximately every **2.50 years**.

Would you like any further details or clarifications?

Here are some related questions to expand on this topic:

1. What would happen to the population growth if the growth rate changed to a different value, such as 1.5?
2. How would the formula change if the initial population were doubled?
3. How can the population model be altered to include a limiting factor, like food availability?
4. How does the logarithmic function help in solving exponential growth equations?
5. What is the importance of understanding doubling time in real-world population studies?

**Tip**: In problems involving exponential growth, the doubling time formula \( t = \frac{\ln(2)}{\ln(1 + r)} \) is useful for quickly calculating how long it takes for any population to double, where \( r \) is the growth rate.

A population of fish is given by P(t) = 3.2 * 10^4 * 1.32^t. When will the population reach 1 million and 1 billion? Find the doubling time of the population that behaves according to the equation P(t) = 1200 * 1.32^t.

Learn how to solve exponential growth problems using logarithms. This example problem involves finding when a fish population reaches 1 million and 1 billion, and determining the doubling time for a population following P(t) = 1200 * 1.32^t. Suitable for high school students in Grades 10-12.

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