https://file.aiquickdraw.com/mathbot/file/1729560980782x4uq226i.jpg

The given function is:

\[
N(t) = N_0 \cdot e^{1.6t}
\]

### 1. Continuous Growth Rate:
The continuous growth rate is the coefficient of \(t\) in the exponent of the expression \(e^{1.6t}\), which is \(1.6\). To convert this into a percentage, we multiply by 100:

\[
1.6 \times 100 = 160\%
\]

So, the continuous growth rate is **160%**.

### 2. Effective Growth Rate:
The effective growth rate can be calculated using the formula:

\[
\text{Effective Growth Rate} = e^r - 1
\]

where \(r = 1.6\). Calculating this:

\[
e^{1.6} \approx 4.953
\]

Thus,

\[
4.953 - 1 = 3.953
\]

Converting this to a percentage gives:

\[
3.953 \times 100 \approx 395\%
\]

So, the effective growth rate is approximately **395%**.

Would you like more details or have any questions?

### Relative Questions:
1. How does continuous growth differ from effective growth in general?
2. What are some real-world examples where continuous growth is applicable?
3. How do you convert a continuous growth rate to an effective growth rate?
4. What is the significance of the base \(e\) in growth equations?
5. Can the continuous growth rate ever be negative, and what would that mean?

### Tip:
When dealing with exponential growth functions, always identify whether you're asked for a continuous rate (using \(r\)) or an effective rate (calculated as \(e^r - 1\)) to avoid confusion.

For the following function, identify the continuous growth rate, then determine the effective growth rate. Assume t is measured in years. N(t) = N0 * e^(1.6t)

Learn how to calculate both the continuous and effective growth rates for an exponential growth function N(t) = N0 * e^(1.6t). This problem focuses on key mathematical concepts such as continuous and effective growth, exponential models, and involves formulas like e^r - 1. Suitable for students in Grades 11-12.

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