https://file.aiquickdraw.com/mathbot/file/1729532388715lni0e7fe.jpg

Let's break down the problem you're working on. There are two parts:

### Part A:
You need to find an **exponential model** for the relationship between **age** \(a\) and **length** \(L(a)\). The data provided are:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Age (years)} & 4 & 8 & 11.5 & 14 & 15 \\
\hline
\text{Length (inches)} & 18 & 24 & 28 & 30 & 32 \\
\hline
\end{array}
\]

The exponential model will have the form:

\[
L(a) = L_0 \cdot e^{ka}
\]

Where:
- \(L_0\) is the initial length (when \(a = 0\)),
- \(k\) is the growth rate,
- \(a\) is the age in years.

To find this model, we typically apply an **exponential regression** on the given data, which you can compute using statistical software (like Excel, TI-84 calculator, or online regression tools).

### Part B:
Once you have the exponential model, you need to calculate the **instantaneous rate of change** of the length when the fish is 4 years old. This rate of change is the **derivative** of the exponential function, evaluated at \(a = 4\).

If \(L(a) = L_0 \cdot e^{ka}\), then the derivative is:

\[
\frac{dL}{da} = L_0 \cdot k \cdot e^{ka}
\]

You will substitute the values from your model into this derivative and evaluate it at \(a = 4\).

Would you like more details on how to do the regression or derivative, or help finding these values directly? Let me know! 

---

Here are 5 questions that can help expand your understanding:
1. What is the process to apply exponential regression on a calculator or in Excel?
2. How do we interpret the parameters \(L_0\) and \(k\) in the context of the fish's growth?
3. Why does the exponential model suit this data, and what does it tell us about the fish's growth pattern?
4. What is the significance of the instantaneous rate of change at a specific age?
5. How would the model change if the growth rate slowed down or sped up significantly?

### Tip:
When performing exponential regression, double-check your inputs for rounding errors, especially since you need to round to the nearest hundredth!

Find an exponential model of the data using an exponential regression where length is a function of age (round all values to the nearest hundredth place). Age (years): [4, 8, 11.5, 14, 15], Length (inches): [18, 24, 28, 30, 32]. Calculate the instantaneous rate of change of the length at the age of 4 years using the rounded model.

Learn how to find an exponential model using regression from given age and length data, and calculate the instantaneous rate of change at a specific age. Data: Age (years) [4, 8, 11.5, 14, 15], Length (inches) [18, 24, 28, 30, 32]. Suitable for high school students in Grades 10-12.

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