91Ó°ÊÓ

In mathematics, a function is a relationship between two sets, typically involving inputs and outputs, where each input is related to exactly one output. In our given problem, we have a function denoted by \( T(L) = 19 - 5 \ln(53 - L) \). This function maps the length \( L \) of a haddock, measured in centimeters, to its age \( T \), measured in years. Here, the length \( L \) is our input, and the age \( T \) is our output. Functions are essential because they provide a systematic way to describe how one quantity depends on another. They can be used to model real-world situations, like determining the age of a haddock based on its length. Whenever you have a function, you can substitute any permissible value of the input (\( L \) in this case) into the function to obtain an output (\( T \)). This relationship is particularly practical for modeling changes and predicting outcomes in various scientific and mathematical fields.

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is an important mathematical function that arises in many areas of science and engineering. In our haddock age problem, the function includes the expression \( \ln(53 - L) \). This represents the natural logarithm of the value \( 53 - L \). The natural logarithm is used in this model because it can effectively describe growth processes, decay processes, and, in this context, how the age of a haddock scales with its length.
Natural logarithms have unique properties:

• \( \ln(1) = 0 \) because any number raised to the power of 0 equals 1.
• \( \ln(e) = 1 \) because the power of \( e \) is itself.
• They convert multiplication into addition, which simplifies complex multiplicative relationships into linear forms.

Using the logarithm in the given function allows us to model nonlinear relationships effectively. Here, it mimics the relationship between the size and age of the haddock, acknowledging that growth rates often decrease over time as captured by the logarithmic scale.

Graphing is a visual way to represent data and understand relationships between variables. In the context of our problem, we are graphing the age \( T \) as a function of length \( L \). This gives us a clear picture of how the age of a haddock varies with its length. To graph the function \( T(L) = 19 - 5 \ln(53 - L) \), follow these steps:

• Choose several values of \( L \) between 25 and 50. These are your x-values.
• Compute \( T(L) \) for each value of \( L \). These are your y-values on the graph.
• Plot each corresponding pair \((L, T)\) as points on a two-dimensional plane.
• Connect these points with a smooth curve to represent the function.

By graphing, you will notice that the age \( T \) changes with the length \( L \) and can identify patterns, such as whether the relationship is increasing or decreasing. The graph can show how the age rises more slowly as the length approaches 50 cm, reflecting a common biological pattern where growth slows as organisms reach fuller maturity.