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### Given:
The function for the length \( L(t) \) of the Clupea harengus (Atlantic herring) as a function of age \( t \) is:

\[
L(t) = 32 \left(1 - e^{-0.37t}\right) \quad \text{cm for} \quad 0 \leq t \leq 13
\]

### Part 1: How fast is the length changing at age \( t = 2 \) years?

To find the rate of change of length, we need to differentiate the function \( L(t) \) with respect to \( t \). The derivative \( \frac{dL}{dt} \) will give the rate of change of length at any age \( t \).

\[
L(t) = 32 \left(1 - e^{-0.37t}\right)
\]

Differentiate with respect to \( t \):

\[
\frac{dL}{dt} = 32 \cdot \frac{d}{dt} \left( 1 - e^{-0.37t} \right) = 32 \cdot \left( 0 - (-0.37)e^{-0.37t} \right)
\]

\[
\frac{dL}{dt} = 32 \cdot 0.37 \cdot e^{-0.37t}
\]

So, the rate of change of length is:

\[
\frac{dL}{dt} = 11.84 e^{-0.37t} \quad \text{cm/year}
\]

Now, substitute \( t = 2 \) into the derivative to find the rate of change at age 2 years:

\[
\frac{dL}{dt} \Big|_{t=2} = 11.84 e^{-0.37(2)}
\]

We will calculate this expression now.The rate of change of length at age \( t = 2 \) years is approximately \( 5.65 \, \text{cm/year} \).

### Part 2: At what age is the length changing at a rate of 6 cm/year?

To solve this, we use the rate of change formula:

\[
\frac{dL}{dt} = 11.84 e^{-0.37t}
\]

We are asked to find \( t \) when \( \frac{dL}{dt} = 6 \). Set up the equation:

\[
6 = 11.84 e^{-0.37t}
\]

Now, solve for \( t \):

\[
e^{-0.37t} = \frac{6}{11.84}
\]

Take the natural logarithm of both sides:

\[
-0.37t = \ln\left(\frac{6}{11.84}\right)
\]

\[
t = \frac{\ln\left(\frac{6}{11.84}\right)}{-0.37}
\]

Let's compute this value.The length of the Atlantic herring is changing at a rate of 6 cm/year when the age is approximately \( 1.84 \) years.

Would you like any further details on these calculations or have any other questions?

Here are 5 related questions you can explore:
1. How does the growth rate change as the herring ages?
2. What is the maximum possible length the herring can reach based on the given model?
3. What is the length of the herring at \( t = 2 \) years?
4. How fast is the length changing at \( t = 5 \) years?
5. What would the rate of change be at \( t = 10 \) years?

**Tip:** In models like this, the growth rate often decreases as the organism approaches its maximum length, which is described by an exponential decay term.

How fast is the length of an Atlantic herring changing at age t = 2 years, and at what age is the length changing at a rate of 6 cm/year?

Learn how to calculate the rate of change of length for an Atlantic herring at age t = 2 years using differentiation and exponential functions. Discover how to determine the age when the length changes at a rate of 6 cm/year.

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