write the function p=100e^0.03t in the form of p=p0a^t

To rewrite the function \( p = 100e^{0.03t} \) in the form \( p = p_0a^t \), we need to express it in terms of a base \( a \) raised to the power of \( t \).

### Given:
\[ p = 100e^{0.03t} \]

We want to express this in the form:
\[ p = p_0 a^t \]

### Step-by-step process:

1. The initial value \( p_0 \) is already present in the original equation, which is 100. Thus, \( p_0 = 100 \).

2. Now, we need to express \( e^{0.03t} \) in the form of \( a^t \).

    Since \( e^{0.03t} \) can be rewritten as \( \left( e^{0.03} \right)^t \), the base \( a \) is \( e^{0.03} \).

Thus, the function becomes:
\[
p = 100 \cdot \left( e^{0.03} \right)^t
\]

### Simplifying further:
The base \( a = e^{0.03} \) is approximately \( 1.03045 \) (since \( e^{0.03} \approx 1.030454533 \)).

Therefore, the equation in the form \( p = p_0 a^t \) is:

\[
p = 100 \cdot (1.03045)^t
\]

This is the function rewritten in the desired form.

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Do you want further details or have any questions?

Here are 5 related questions:
1. What is the significance of the base \( a \) in exponential growth?
2. How would the equation change if the growth rate was 0.05 instead of 0.03?
3. How can this function be used to model compound interest over time?
4. What is the value of \( p \) after 10 years using the original equation?
5. How do you interpret \( p_0 \) in a real-world application?

**Tip**: The number \( e \) (Euler's number) is commonly used in continuous growth problems, while base \( a \) is often used in situations of compounded growth.

Learn how to rewrite the exponential growth function p = 100e^0.03t in the form of p = p_0 a^t, where the base a is derived from the growth rate. This solution includes the key formula p = p_0 a^t, simplifying e^0.03 into the approximate base 1.03045. Suitable for Grades 10-12.

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