Problem 98 - Algebra and Trigonometry with Analytic Geometry

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 weight of a 25-foot sei whale is approximately 8.46 tons.

Step 1: Understand the Problem

We are given the relationship between the whale's length and weight through the equation \( W = 0.0016 L^{2.43}\). We need to find the weight \(W\) of a sei whale given its length \(L = 25\) feet.

Step 2: Substitute the Length into the Equation

Substitute \(L = 25\) into the equation \( W = 0.0016 L^{2.43}\): \[ W = 0.0016 \times 25^{2.43} \]

Step 3: Calculate the Power

Calculate \(25^{2.43}\). Using a calculator, find the value of \(25^{2.43}\), which is approximately \(5289.47\).

Step 4: Calculate the Weight

Now, multiply \(0.0016\) by the value obtained from the previous step: \[ W = 0.0016 \times 5289.47 \] This gives us approximately \(W = 8.46\) tons.

Step 5: Conclude the Solution

Thus, the estimated weight of a sei whale that is 25 feet long is approximately 8.46 tons.

Key Concepts

Exponential Functions

Exponential functions involve equations where the variable is raised to a power. These are integral in various fields, including biology, finance, and physics. In our given example, the formula \( W = 0.0016 L^{2.43} \) helps us estimate the weight of a sei whale based on its length. The variable \( L \) is raised to an exponent of \( 2.43 \), which means it becomes an exponential function. This kind of function is useful when growth is not simply linear. For example:

• The areas and volumes often scale with powers of dimensions, reflecting real-world phenomena more accurately.
• Growth patterns in populations can be modeled exponentially, as can decay processes.

When dealing with such functions, it's important to use precise mathematical tools like calculators or software to compute the powers correctly. This ensures we capture the significant exponential effects accurately.

Equation Solving

Equation solving is the process of finding the value of a variable that makes the equation true. In basic algebra, this might involve solving a simple linear equation. However, as equations become more complex, especially with exponential terms, the process requires careful manipulation.
For the sei whale problem, solving the equation:

• We first substitute the known variable, the whale's length \( L = 25 \), into the formula \( W = 0.0016 L^{2.43} \).
• Following that, solve for the unknown \( W \), which is the whale's weight.

Equation solving sometimes needs additional techniques, such as factoring, using the quadratic formula, or graphing. In our case, computational tools help in dealing with the power function. Understanding how to solve these equations allows us to interpret and utilize mathematical models effectively.

Substitution Method

The substitution method is a straightforward technique used in algebra to simplify problems by replacing variables with known values. This is particularly useful in solving equations or systems of equations.
In our whale weight problem, the substitution method is applied early in the steps:

• Identify the variable to substitute; here, it is the length \( L \) of the whale.
• Replace \( L \) with 25 in the equation to directly compute the effect on \( W \).

The advantage of substitution is that it reduces the complexity of equations by transforming them into simpler arithmetic calculations. Therefore, it builds a direct link between theory (equation) and practical calculation (numerical estimate). This method is a valuable tool in solving not only algebraic problems but also in calculus and other areas of mathematics.