Problem 109 - A Graphical Approach to College Algebra

Question

The number of species in a sample is approximated by \(S(n)=a \ln \left(1+\frac{n}{a}\right),\) where \(n\) is the number of individuals in the sample and \(a\) is a constant that indicates the diversity of species in the community. If \(a=0.36,\) find \(S(n)\) for each value of \(n\) (Hint: \(S(n)\) must be a whole number.) (a) 100 (b) 200 (c) 150 (d) 10

Step-by-Step Solution

Answer

S(100) ≈ 2, S(200) ≈ 3, S(150) ≈ 3, S(10) ≈ 1.

Step 1: Substitute the Value of 'a' into the Formula

The formula provided for the number of species is \( S(n) = a \ln\left( 1 + \frac{n}{a} \right) \). We are given that \( a = 0.36 \). Substitute this value into the formula to get \( S(n) = 0.36 \ln\left( 1 + \frac{n}{0.36} \right) \).

Step 2: Calculate S(n) for n = 100

Substitute \( n = 100 \) into the formula: \[ S(100) = 0.36 \ln\left( 1 + \frac{100}{0.36} \right) = 0.36 \ln(278) \]. Calculate \( \ln(278) \) using a calculator and round the result to the nearest whole number.

Step 3: Calculate S(n) for n = 200

Substitute \( n = 200 \) into the formula: \[ S(200) = 0.36 \ln\left( 1 + \frac{200}{0.36} \right) = 0.36 \ln(556) \]. Calculate \( \ln(556) \) using a calculator and round the result to the nearest whole number.

Step 4: Calculate S(n) for n = 150

Substitute \( n = 150 \) into the formula: \[ S(150) = 0.36 \ln\left( 1 + \frac{150}{0.36} \right) = 0.36 \ln(417) \]. Calculate \( \ln(417) \) using a calculator and round the result to the nearest whole number.

Step 5: Calculate S(n) for n = 10

Substitute \( n = 10 \) into the formula: \[ S(10) = 0.36 \ln\left( 1 + \frac{10}{0.36} \right) = 0.36 \ln(28.78) \]. Calculate \( \ln(28.78) \) using a calculator and round the result to the nearest whole number.

Key Concepts

Natural Logarithm

Understanding the natural logarithm is crucial in the species diversity formula. The natural logarithm, denoted as \( \ln \), is the logarithm to the base of the mathematical constant \( e \), approximately equal to 2.718. Unlike common logarithms that use base 10, natural logarithms are particularly useful in scientific calculations because they simplify the mathematical treatment of continuous growth processes.
For example, in biology and ecology, growth rates or decay rates often follow an exponential function, making the natural logarithm a fitting choice. In the species diversity formula, the term \( \ln(1 + \frac{n}{a}) \) represents the natural logarithm of a calculated ratio based on the sample size \( n \) and the constant \( a \). This ratio helps determine the complexity of a community, as a slight change in the number of individuals \( n \) leads to logarithmic changes in the perceived number of species.
Additionally, the natural logarithm has unique properties like the fact that \( \ln(1) = 0 \) and it approaches infinity as \( x \) increases, which are valuable when dealing with large sample sizes in population studies.

Sample Size Calculation

Sample size calculation is key to estimating species diversity. In our exercise, the sample size \( n \) indicates the number of individuals in a community population being studied. Calculations are done using the formula \( S(n) = a \ln(1 + \frac{n}{a}) \).
Here, \( n \) directly affects the perceived number of species \( S(n) \). As you increase the sample size, \( \frac{n}{a} \) also increases, leading to a higher value inside the logarithm. This, in turn, increases the calculated species number. For instance, in our calculations:

• For \( n = 100 \), we calculate \( 0.36 \ln(278) \).
• For \( n = 200 \), this changes to \( 0.36 \ln(556) \), clearly showing that as the sample size doubles, it affects \( S(n) \).


• Smaller sample sizes, such as \( n = 10 \), provide smaller ratios, i.e., \( 0.36 \ln(28.78) \), limiting the potential to fully understand the species diversity.

This highlights why careful selection of \( n \) is essential, especially for ecological diversity studies above sample sizes.

Mathematical Approximation

Mathematical approximation is a technique used to simplify complex calculations. In the context of the given formula for species diversity, it's necessary to ensure the results are practical and workable. Calculating \( \ln \) values can result in irrational numbers; hence, approximation is critical to arrive at whole number results as required.
For example, when calculating \( S(n) \) for \( n = 100 \), it's important to initially derive an accurate logarithmic value for \( \ln(278) \). However, this value often can't be expressed as a clean decimal, let alone a whole number. Therefore, rounding off to the nearest whole number becomes essential.
Subsequently, mathematical approximations ensure that results like \( S(100) \), \( S(200) \), etc., are useful and readily interpretable, important for making informed ecological observations and conclusions. Approximations determine how finer or broader predictions might be, impacting how species data is interpreted and utilized for scientific studies.