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.