Problem 98 - Precalculus: Functions and Graphs

Question

The length-weight relationship for the sei whale can be approximated by \(W=0.0016 L^{2.43},\) where \(W\) is in tons and \(L\) is in feet. Estimate the weight of a whale that is 25 feet long.

Step-by-Step Solution

Answer

The whale weighs approximately 0.93 tons.

Step 1: Identify Known Values

We need to determine the weight, \(W\), of a sei whale that is 25 feet long. Given in the problem, \(L = 25\) feet.

Step 2: Use the Relationship Formula

The relationship between the length \(L\) and weight \(W\) of the sei whale is given by the formula \(W = 0.0016 L^{2.43}\).

Step 3: Substitute Length into Formula

Substitute the known length \(L = 25\) into the equation to find the weight: \[ W = 0.0016 imes 25^{2.43} \]

Step 4: Calculate the Exponent

Calculate \(25^{2.43}\) first. This yields approximately \(582.939\).

Step 5: Calculate the Product

Substitute the exponent value back into the formula: \[ W = 0.0016 imes 582.939 \] This results in \(W \approx 0.9327\).

Step 6: Interpret the Result

The calculated weight \(W\) is approximately \(0.9327\) tons, which suggests the whale weighs just under one ton.

Key Concepts

Mathematical Formulas

Mathematical formulas are expressions that form the foundation of many scientific and engineering calculations. In this context, we explore a specific formula used to approximate the relationship between the length and weight of sei whales. The formula used is: \[ W = 0.0016 L^{2.43} \] Here:

• \(W\) represents the weight of the whale in tons.
• \(L\) is the whale's length in feet.

This exponential formula indicates that as the length of the whale increases, its weight increases at a rate determined by the power 2.43. The use of mathematical formulas like this enables us to make quick and practical predictions based on known variables. We convert the unknown length into a calculable weight by plugging in the length measurement into the formula and solving the equation.

Substitution Method

The substitution method involves taking known values and "substituting" them into a given formula to find an unknown variable. Let's delve into how this method is applied in our whale weight problem: First, identify the known values. In our example, we know the length \(L\) of the whale is 25 feet. By substituting this value into our formula: \[ W = 0.0016 \times (25)^{2.43} \] we replace \(L\) with the given length. This step is crucial as it allows us to transition from the formula stage to actually computing an answer. After substitution, the problem becomes a straightforward series of arithmetic operations.

Mathematical Modeling

Mathematical modeling is a powerful technique used to represent real-world phenomena with mathematical expressions. It helps simplify complex concepts by creating a visual or quantitative framework that can be analyzed and manipulated. In this case, the relationship between the sei whale's length and weight is an excellent example of mathematical modeling.

• The model \(W = 0.0016 L^{2.43}\) provides a simplified way to predict a whale’s weight based solely on its length.
• This model is derived from scientific observations, where data is collected and analyzed to form a generalized equation that can be applied to specific scenarios.

Using such a model, where complex biological factors are encapsulated into a manageable equation, allows for quick and easy problem-solving without the need for extensive measurement or data collection each time. This approach not only saves time but also enhances our understanding and provides insights that contribute to further scientific studies and ecological assessments.