Enter the function P=13e^0.43t in the form P=P_0a^t Does it represent exponential growth or exponential decay Round base of the exponential, a, to four decimal places



The given function is \( P = 13e^{0.43t} \). To express this in the form \( P = P_0 a^t \), follow these steps:

1. Identify \( P_0 \): In this form, \( P_0 \) is the initial value when \( t = 0 \). In this case, \( P_0 = 13 \).

2. Convert the base \( e \) to an exponential form: 
   \[
   P = 13e^{0.43t} \quad \text{can be rewritten as} \quad P = 13\left(e^{0.43}\right)^t
   \]
   Now, calculate \( e^{0.43} \) to four decimal places.

\[
e^{0.43} \approx 1.5374
\]

So the function becomes:
\[
P = 13(1.5374)^t
\]

### Exponential Growth or Decay:
Since \( a = 1.5374 \), which is greater than 1, the function represents **exponential growth**.

### Final Form:
\[
P = 13(1.5374)^t
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. What is the value of \( P \) when \( t = 2 \)?
2. How does the value of \( P_0 \) affect the overall growth?
3. What would the function look like if it represented exponential decay?
4. How do you calculate the doubling time for this growth model?
5. How does the constant \( e \) relate to exponential growth functions?

**Tip:** In exponential functions, if the base \( a > 1 \), it signifies growth, while \( 0 < a < 1 \) indicates decay.

Enter the function P=13e^0.43t in the form P=P_0a^t. Does it represent exponential growth or exponential decay? Round base of the exponential, a, to four decimal places.

Explore how to convert the function P=13e^0.43t into the exponential form P=P_0a^t and determine if it represents growth or decay. This problem covers exponential functions and includes rounding the base 'a' to four decimal places. Ideal for Grades 10-12 students.

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