Find the exponential function of the form A(t)=Ie^rt using a log transformation for the points (5, 250) and (9, 1360)

To find the exponential function of the form \( A(t) = Ie^{rt} \) that passes through the points \((5, 250)\) and \((9, 1360)\), we can follow these steps:

1. **Set up the equations:**

   The general form of the exponential function is:
   \[
   A(t) = Ie^{rt}
   \]

   For the point \((5, 250)\):
   \[
   250 = Ie^{5r}
   \]

   For the point \((9, 1360)\):
   \[
   1360 = Ie^{9r}
   \]

2. **Solve the equations using logarithms:**

   To eliminate \( I \), we can divide the second equation by the first equation:
   \[
   \frac{1360}{250} = \frac{Ie^{9r}}{Ie^{5r}}
   \]

   Simplify the equation:
   \[
   \frac{1360}{250} = e^{4r}
   \]

   \[
   \frac{1360}{250} = 5.44
   \]

   \[
   5.44 = e^{4r}
   \]

3. **Solve for \( r \):**

   Take the natural logarithm of both sides:
   \[
   \ln(5.44) = \ln(e^{4r})
   \]

   \[
   \ln(5.44) = 4r
   \]

   \[
   r = \frac{\ln(5.44)}{4}
   \]

   Calculate the value:
   \[
   r \approx \frac{1.695}{4}
   \]

   \[
   r \approx 0.42375
   \]

4. **Find \( I \):**

   Use the value of \( r \) in one of the original equations to find \( I \). Let's use the first equation \((5, 250)\):
   \[
   250 = Ie^{5 \cdot 0.42375}
   \]

   \[
   250 = Ie^{2.11875}
   \]

   \[
   250 = I \cdot 8.318
   \]

   \[
   I \approx \frac{250}{8.318}
   \]

   \[
   I \approx 30.05
   \]

So, the exponential function is:
\[
A(t) = 30.05e^{0.42375t}
\]

Would you like more details or have any questions? Here are some further questions you might consider:

1. How do you determine the initial value \( I \) in an exponential function?
2. What is the significance of the base \( e \) in exponential functions?
3. How can you verify that the function fits the given points?
4. How does changing \( r \) affect the shape of the exponential graph?
5. Can exponential functions be used to model all types of growth? Why or why not?

**Tip:** Always verify your exponential model by substituting the given points back into the equation to ensure accuracy.

Learn how to find the exponential function A(t)=Ie^rt passing through the points (5, 250) and (9, 1360) using logarithms. Understand the steps involving logarithmic transformations and calculations for r and I. Suitable for advanced high school students studying exponential functions.

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