The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population 
P0
 has doubled in 5 years.
Suppose it is known that the population is 8,000 after 3 years. What was the initial population P0? (Round your answer to one decimal place.)
P0 = 
 
Incorrect: Your answer is incorrect.
What will be the population (in persons) in 10 years? (Round your answer to the nearest person.)
 
Incorrect: Your answer is incorrect.
 persons
How fast (in persons/year) is the population growing at 
t = 10?
 (Round your answer to the nearest person.)
 
Incorrect: Your answer is incorrect.
 persons/year


This problem involves exponential population growth, where the population grows at a rate proportional to the number of people present. The mathematical model for exponential growth is:

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

Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population (at \( t = 0 \)),
- \( k \) is the growth rate constant,
- \( t \) is the time in years.

### 1. Finding the initial population \( P_0 \)
We are given two pieces of information:
1. The population doubles in 5 years. This implies \( P(5) = 2P_0 \).
2. The population is 8,000 after 3 years, i.e., \( P(3) = 8000 \).

Let's first use the second piece of information to express the population after 3 years:
\[
P(3) = P_0 e^{3k} = 8000
\]

Now, from the first piece of information, we know that the population doubles in 5 years, so:
\[
P(5) = P_0 e^{5k} = 2P_0
\]

We can divide these two equations to eliminate \( P_0 \) and solve for \( k \):

\[
\frac{P(5)}{P(3)} = \frac{P_0 e^{5k}}{P_0 e^{3k}} = \frac{2P_0}{8000}
\]

This simplifies to:

\[
e^{2k} = \frac{2P_0}{8000}
\]

Solving for

The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population P0 has doubled in 5 years. Suppose it is known that the population is 8,000 after 3 years. What was the initial population P0? (Round your answer to one decimal place.) What will be the population (in persons) in 10 years? (Round your answer to the nearest person.) How fast (in persons/year) is the population growing at t = 10? (Round your answer to the nearest person.)

Solve an exponential growth problem where a population doubles in 5 years and reaches 8,000 in 3 years. Learn how to find the initial population, future population at 10 years, and the growth rate at t = 10. This step-by-step solution is ideal for high school students.

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