Problem 67 - A Graphical Approach to College Algebra

Question

Fiddler Crab Size One study of the male fiddler crab showed a connection between the weight of its claws and the animal's total body weight. For a crab weighing over 0.75 gram, the weight of its claws can be estimated by $$ f(x)=0.445 x^{1.25} $$ The input \(x\) is the weight of the crab in grams, and the output \(f(x)\) is the weight of the claws in grams. Predict the weight of the claws for a crab that weighs 2 grams. (Source: Huxley, J., Problems of Relative Growth, Methuen and Co.; Brown, D. and P. Rothery, Models in Biology: Mathematics, Statistics, and Computing, John Wiley and Sons.)

Step-by-Step Solution

Answer

The claw weight is about 1.058 grams.

Step 1: Understand the Function

The function provided is \( f(x) = 0.445 x^{1.25} \). This function will help us estimate the weight of the crab's claws based on the total weight of the crab.

Step 2: Identify the Input Value

We are given that a particular fiddler crab weighs 2 grams. Therefore, in this exercise, the input value \( x \) is 2 grams.

Step 3: Plug the Input into the Function

Insert the given weight of the crab, 2 grams, into the function: \( f(2) = 0.445 \times 2^{1.25} \).

Step 4: Calculate the Exponent

Calculate \( 2^{1.25} \). Begin by finding the square root of 2 (which is \( 2^{0.5} \)) and then multiply it by 2. This can be done using a calculator as follows: \( 2^{1.25} \approx 2.378 \).

Step 5: Multiply by the Coefficient

Multiply the result from Step 4 by the coefficient in the function: \( 0.445 \times 2.378 \approx 1.058 \).

Step 6: Conclude with the Predicted Claw Weight

The estimated weight of the crab's claws is approximately 1.058 grams.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In our fiddler crab example, the exponentiation is represented as \( x^{1.25} \), where the base is the weight of the crab in grams \( x \), and the exponent is 1.25. This operation is crucial in the formula used to predict the weight of the crab's claws based on its overall weight. When performing exponentiation, you raise the base (in this case 2 grams) to the power of the exponent (1.25).

Exponents like 1.25 mean the base number is to be multiplied recursively. In simpler terms:

• The exponent is split into whole and fractional parts (1 becomes multiplying by 2, and 0.25 is part of a more detailed root calculation).
• Specifically, \(2^{1.25}\) translates to \((2^{1} \times 2^{0.25})\) which equals multiplying the number by itself once and then taking it to the quarter power.

Using a calculator often helps in handling such operations with non-integer exponents, ensuring accuracy. Exponentiation allows us to model certain biological relationships such as the claw weight concerning the total weight of the crab with precision.

Mathematical Modeling

Mathematical modeling is a method of representing real-world scenarios using mathematical expressions. In the exercise, we employ a model with the function \( f(x) = 0.445 x^{1.25} \) to predict the weight of the claws of a fiddler crab. This model reflects the biological observation that the claw weight changes in relation to the overall weight based on a power law relationship.

In this context:

• The constant 0.445 is a coefficient that adjusts the scaling of the function to match empirical data.
• The term \( x^{1.25} \) signifies the rate at which the claw weight grows as a function of body weight. Exponents in such models often represent nuanced biological growth rates or scaling laws.

Through mathematical modeling, complex biological relationships can be simplified into manageable computations, enabling predictions and insights to be drawn from existing data. It's a powerful tool in fields like biology where measurements need precise interpretation.

Step-by-Step Calculations

Step-by-step calculations are essential for understanding how a mathematical model is applied to real data. Let's break down the process as seen in the fiddler crab claw weight estimate.

• Step 1: Understand the Function - Recognize the purpose of the function \( f(x) = 0.445 x^{1.25} \). It estimates claw weight based on the crab's total weight.
• Step 2: Identify the Input Value - For a crab weighing 2 grams, identify this weight as the input value \( x \).
• Step 3: Plug the Input into the Function - Substitute 2 grams into the equation: \( f(2) = 0.445 \times 2^{1.25} \).
• Step 4: Calculate the Exponent - Compute \( 2^{1.25} \). This involves calculating the fractional exponent by manipulating it into a series of roots and powers, approximately yielding 2.378.
• Step 5: Multiply by the Coefficient - Multiply the calculated exponentiated value by 0.445 to achieve the final output as \( 1.058 \) grams.

By analyzing each step carefully, we can understand how the model functions and ensures accuracy in drawing conclusions about scientific observations. Following each stage meticulously leads to clearer understanding and confidence in using mathematical models.