
Find the exponential function of the form A(t)=Ie^rt using a log transformation for the points (4, 120) and (10, 35)


To find the exponential function of the form \( A(t) = Ie^{rt} \) that passes through the points \((4, 120)\) and \((10, 35)\), we can use logarithms as follows:

1. **Set up the equations:**

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

   For the point \((4, 120)\):
   \[
   120 = Ie^{4r}
   \]

   For the point \((10, 35)\):
   \[
   35 = Ie^{10r}
   \]

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

   To eliminate \( I \), we can divide the second equation by the first equation:
   \[
   \frac{35}{120} = \frac{Ie^{10r}}{Ie^{4r}}
   \]

   Simplify the equation:
   \[
   \frac{35}{120} = e^{10r - 4r}
   \]

   \[
   \frac{35}{120} = e^{6r}
   \]

   Simplify further:
   \[
   \frac{35}{120} = \frac{7}{24}
   \]

   \[
   \frac{7}{24} = e^{6r}
   \]

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

   Take the natural logarithm of both sides:
   \[
   \ln\left(\frac{7}{24}\right) = \ln(e^{6r})
   \]

   \[
   \ln\left(\frac{7}{24}\right) = 6r
   \]

   \[
   r = \frac{\ln\left(\frac{7}{24}\right)}{6}
   \]

   Calculate the value:
   \[
   r \approx \frac{\ln(7) - \ln(24)}{6}
   \]

   \[
   r \approx \frac{1.9459 - 3.1781}{6}
   \]

   \[
   r \approx \frac{-1.2322}{6}
   \]

   \[
   r \approx -0.20537
   \]

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

   Use the value of \( r \) in one of the original equations to find \( I \). Let's use the first equation \((4, 120)\):
   \[
   120 = Ie^{4 \cdot -0.20537}
   \]

   \[
   120 = Ie^{-0.82148}
   \]

   \[
   120 = I \cdot 0.43991
   \]

   \[
   I \approx \frac{120}{0.43991}
   \]

   \[
   I \approx 272.76
   \]

So, the exponential function is:
\[
A(t) = 272.76e^{-0.20537t}
\]

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

1. How can you confirm that the function fits the given points?
2. What does a negative \( r \) indicate about the behavior of the function?
3. How can you apply this method to more than two points?
4. How does the initial value \( I \) affect the graph of the function?
5. What are some real-life applications of exponential decay functions?

**Tip:** When solving for \( r \) using logarithms, ensure you apply properties of logarithms correctly to simplify the process.

Learn how to find the exponential function A(t) = Ie^rt using a log transformation method with detailed step-by-step instructions. Solve for parameters I and r using points (4, 120) and (10, 35).

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