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To solve this problem, we need to approach it in two parts:

### Part A: Exponential Regression
The goal is to find an exponential model of the data where the length \( L(a) \) is a function of age \( a \), using exponential regression. The general form of an exponential function is:
\[
L(a) = L_0 \cdot e^{k \cdot a}
\]
where:
- \( L_0 \) is the initial value (length when age is 0),
- \( k \) is the rate constant (which we need to find),
- \( a \) is the age in years,
- \( L(a) \) is the length in inches.

The given data points are:
\[
\text{Age} \ (a): \ 4, \ 8, \ 11.5, \ 14, \ 15
\]
\[
\text{Length} \ (L): \ 18, \ 24, \ 28, \ 30, \ 32
\]
We will use exponential regression techniques (e.g., fitting the model using a calculator or a tool like Excel or a graphing calculator) to obtain the parameters \( L_0 \) and \( k \).

### Part B: Instantaneous Rate of Change
After finding the exponential model \( L(a) \), the instantaneous rate of change at age 4 can be calculated by taking the derivative of \( L(a) \) and evaluating it at \( a = 4 \). The derivative of the exponential function is:
\[
\frac{dL}{da} = L_0 \cdot k \cdot e^{k \cdot a}
\]
Substitute the values from the model to compute the rate at age 4.

I can help you solve the regression numerically if needed or explain the steps further. Would you like more details on the regression process or help with calculations?

### 5 Related Questions:
1. How do you compute exponential regression using a calculator?
2. What is the meaning of the derivative in the context of this problem?
3. How do you determine the growth rate \( k \) from data points in an exponential model?
4. What are some examples of exponential growth in biology besides fish length?
5. How can you approximate the instantaneous rate of change for non-exponential models?

### Tip:
When working with exponential models, always check if your data fits the exponential form by plotting it on a semi-logarithmic graph—this will help confirm if exponential regression is appropriate.

Find an exponential model of the data using an exponential regression where length is a function of age (round all values to the nearest hundredths place): Age (years): 4, 8, 11.5, 14, 15 Length (inches): 18, 24, 28, 30, 32

Learn how to use exponential regression to model the length of a fish as a function of age, and calculate the instantaneous rate of change at specific ages. This problem includes detailed data points and teaches concepts like exponential growth and derivatives, suitable for high school students.

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