Suppose P = f(t) is the population (in thousands) of town t years after 1990, and that f(5)=13 and f(9)=18, (a)) Find a formula for f(t) assuming f is exponential: P=f(t)= ?


Given that \( f(t) \) is exponential, we can assume a general exponential form:

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

where:
- \( P_0 \) is the initial population at \( t = 0 \),
- \( k \) is the growth rate.

### Step 1: Set up the equations

From the given information:
1. \( f(5) = 13 \), which implies:
   \[
   13 = P_0 \cdot e^{5k}
   \]
2. \( f(9) = 18 \), which implies:
   \[
   18 = P_0 \cdot e^{9k}
   \]

### Step 2: Divide the equations to solve for \( k \)

To eliminate \( P_0 \), we can divide the second equation by the first:

\[
\frac{18}{13} = \frac{P_0 \cdot e^{9k}}{P_0 \cdot e^{5k}}
\]

This simplifies to:

\[
\frac{18}{13} = e^{4k}
\]

### Step 3: Solve for \( k \)

Take the natural logarithm on both sides:

\[
\ln \left(\frac{18}{13}\right) = 4k
\]

Then:

\[
k = \frac{\ln \left(\frac{18}{13}\right)}{4}
\]

Calculating this value:

\[
k \approx \frac{\ln(1.3846)}{4} \approx \frac{0.325}{4} \approx 0.08125
\]

### Step 4: Solve for \( P_0 \)

Now that we have \( k \), we can substitute it back into one of the original equations to find \( P_0 \). Using \( f(5) = 13 \):

\[
13 = P_0 \cdot e^{5 \cdot 0.08125}
\]

Calculating the exponential term:

\[
13 = P_0 \cdot e^{0.40625} \approx P_0 \cdot 1.501
\]

Then:

\[
P_0 \approx \frac{13}{1.501} \approx 8.66
\]

### Final Answer

The formula for \( f(t) \), the population \( t \) years after 1990, is approximately:

\[
f(t) = 8.66 \cdot e^{0.08125 \cdot t}
\]

This gives us the model for the population growth of the town.

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Would you like a deeper breakdown or to verify each calculation?

### Related Questions

1. How would the model change if \( f \) were assumed to be a linear function instead of exponential?
2. What would the population be in the year 2000 according to this model?
3. How does changing the value of \( k \) affect the growth rate of \( f(t) \)?
4. Can this model be used for long-term predictions, say in 100 years? Why or why not?
5. How would we determine the half-life of the population if it were declining exponentially?

### Tip
When dealing with exponential functions, dividing two instances at different times can often simplify solving for the growth rate \( k \) without needing to know the initial population.

Suppose P = f(t) is the population (in thousands) of town t years after 1990, and that f(5) = 13 and f(9) = 18. (a) Find a formula for f(t) assuming f is exponential: P = f(t) = ?

Learn how to determine an exponential growth formula for a population model. Given f(5) = 13 and f(9) = 18, we derive the growth rate and initial population to solve P = f(t). Suitable for Grades 10-12.

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