Problem 97 - Algebra and Trigonometry with Analytic Geometry

Question

The length-weight relationship for Pacific halibut can be approximated by the formula \(L=0.46 \sqrt[3]{W}\), where \(W\) is in kilograms and \(L\) is in meters. The largest documented halibut weighed 230 kilograms. Estimate its length.

Step-by-Step Solution

Answer

The estimated length of the halibut is approximately 2.82 meters.

Step 1: Understanding the Formula

The formula given is \(L = 0.46 \sqrt[3]{W}\), where \(L\) is the length in meters and \(W\) is the weight in kilograms. Our goal is to estimate the length of a halibut that weighs 230 kilograms.

Step 2: Substitution of Known Values

Substitute \(W = 230\) into the formula. This gives us \(L = 0.46 \sqrt[3]{230}\).

Step 3: Cube Root Calculation

Calculate the cube root of 230. This is \(\sqrt[3]{230} \approx 6.13098\).

Step 4: Multiply by Constant

Multiply the result of the cube root by 0.46: \(L = 0.46 \times 6.13098 \approx 2.82125\).

Step 5: Round the Result

Round the result to an appropriate number of decimal places for clarity. The estimated length is approximately \(2.82\) meters.

Key Concepts

Cube Root Calculation

Calculating the cube root is a key step when solving problems where cube roots are involved, such as estimating the length of a Pacific halibut based on its weight. In mathematics, the cube root of a number, denoted as \( \sqrt[3]{x} \), is a special number that, when multiplied by itself three times, will give the original number. For instance, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).

To calculate the cube root:

• First, understand the method you will use, be it a calculator, guess-and-check, or another technique.
• For large or complex numbers, a calculator is often the most efficient tool.
• Input the number for which you need the cube root, here it is 230, into the cube root function on your calculator.

Executing these steps will reveal that the cube root of 230 is approximately 6.13098. This result is crucial in the formula \(L = 0.46 \sqrt[3]{W}\) used for determining the length of the halibut. By understanding how to perform cube root calculations, students can easily apply this skill to various algebraic formulas.

Length-Weight Relationship

The length-weight relationship in biological contexts, such as with halibut, is essential for estimating the size of a fish from its weight without the need for physical measurement. This relationship is an example of a mathematical model used to describe how certain physical characteristics relate to each other.

In our formula, \(L = 0.46 \sqrt[3]{W}\), we use the cube root of the weight to approximate the length. The cube root function suggests that as the weight increases, the length increases at a decreasing rate. This makes sense biologically because a heavier fish does not grow linearly in terms of length. Instead, its growth tapers as it becomes bulkier rather than longer.

Key points include:

• Constant factor: The constant 0.46 scales the relationship so that the calculated length is realistic concerning empirical data.
• Real-life applications: Fishermen and researchers can use this relationship to make quick length estimates in the field.

Understanding these relationships help in predicting physical traits and managing fish populations efficiently.

Substitution Method

The substitution method is a fundamental approach in algebra used to solve equations where one variable is expressed in terms of another. It involves replacing a variable with its known value to find an unknown value in the equation.

Here's how the substitution method works:

• Identify known values: In the given formula \(L = 0.46 \sqrt[3]{W}\), we know the weight \(W\) of the halibut, which is 230 kilograms.
• Substitute known values: Replace \(W\) in the equation with 230, simplifying the equation to \(L = 0.46 \sqrt[3]{230}\).
• Solve for the unknown: Calculate the cube root of 230, and multiply by 0.46 to find the halibut's length.

By substituting known values into equations, complex problems become more manageable. This method is particularly beneficial in solving real-world problems where measurements or values are subject to formulas, allowing for quick and efficient problem-solving.