Problem 56 - Algebra 1

Question

Use the following information. The average weight \(w\) (in pounds) of an Atlantic cod \(t\) years old can be modeled by the equation \(w=1.16(1.44)^{t} .\) (image cannot copy) According to the model, what is the average weight of an Atlantic cod when it is hatched? How did you get your answer?

Step-by-Step Solution

Answer

The average weight of an Atlantic cod when it is hatched, i.e., when \(t=0\), is 1.16 pounds.

Step 1: Understanding the problem

The problem requires calculation of the average weight for an Atlantic cod at the hatching stage, i.e., when it is 0 years old. Given the formula \(w=1.16(1.44)^{t}\), we need to substitute \(t=0\) in the given equation to find the value of \(w\).

Step 2: Substitution

Substitute the value of \(t=0\) in the given equation, the equation becomes \(w=1.16(1.44)^{0}\). Now, any number to the power of zero except zero is 1. Thus, our equation simplifies to \(w=1.16*1\).

Step 3: Calculate w

Calculate the value for \(w\). Thus, the weight \(w\) in pounds becomes 1.16.

Key Concepts

Modeling Equations

Modeling equations are a powerful tool used to represent real-life situations mathematically. In this scenario, the equation \( w = 1.16(1.44)^{t} \) models the growth in the average weight of an Atlantic cod as it ages. Each part of this equation has a specific role:

• Initial Value (1.16): This is the starting point, representing the average weight of the cod when it is first hatched.
• Growth Factor (1.44): This number reveals how the weight increases over each passing year. It indicates a growth per year basis, portraying exponential growth.
• Exponent (t): Represents the number of years since the cod hatched. This variable allows us to calculate the weight for any given age \( t \).

Understanding modeling equations helps in predicting future outcomes, such as estimating biological growth patterns or financial forecasts. In our equation, as the cod ages, the exponential function allows us to estimate its weight at different times.

Substitution Method

The substitution method is crucial in solving equations like the one in this exercise. It involves replacing a variable with a specific value to evaluate or simplify an equation. Here’s a brief breakdown of how it’s used:

• Select a Variable: In the equation \( w = 1.16(1.44)^{t} \), we focus on the variable \( t \).
• Substitute the Value: To find the weight of the cod when hatched, substitute \( t = 0 \). This yields \( w = 1.16(1.44)^{0} \).
• Simplify the Equation: Remember, any number raised to the power of zero is 1 (except zero itself), so \( (1.44)^{0} = 1 \). Therefore, the equation simplifies to \( w = 1.16 \times 1 \).

This method is exceptionally useful for determining values at specific instances, like when a system first begins or is unaltered, as exemplified here with age \( t = 0 \). It ensures accuracy in early stages of growth or processes.

Age-Related Calculations

Age-related calculations are typically used in various fields such as biology to understand growth patterns over time. Using equations to calculate age-specific variables helps us interpret the growth milestones of organisms or systems. With the equation \( w = 1.16(1.44)^{t} \), you can determine the weight of an Atlantic cod for any age \( t \). Here's why this is important:

• Prediction: Helps predict future growth. By inputting different ages, you can create a growth chart for the cod.
• Comparison: Allows comparison between actual growth and the modeled prediction to assess accuracy.
• Conservation: Understanding natural growth rates can aid in the species' sustainable management and conservation efforts.

By learning to utilize this equation for various ages, you can effectively map out how an organism or variable changes over time, crucial for studies and applications involving life cycles and developmental biology.