Problem 60 - Algebra 1: Concepts and Skills

Question

Use the example on the previous page as a model. The average weight \(w\) (in pounds) of an Atlantic cod can be modeled by \(w=1.21(1.42)^{t}\) where \(t\) is the age of the fish (in years). Find the ratio of the weight of a 5 -year-old cod to the weight of a 2 -year-old cod.

Step-by-Step Solution

Answer

The ratio of the weight of a 5-year-old cod to a 2-year-old cod, according to the given model, is \((1.42)^{3}\)

Step 1: Calculate the weight of a 5-year-old cod

Firstly, substitute \(t = 5\) into the equation to get the weight of a 5-year-old cod: \(w_{5}=1.21(1.42)^{5}\)

Step 2: Calculate the weight of a 2-year-old cod

Secondly, substitute \(t = 2\) into the equation to get the weight of a 2-year-old cod: \(w_{2}=1.21(1.42)^{2}\)

Step 3: Compute the ratio

The ratio of the weight of a 5-year-old cod to a 2-year-old cod is calculated by dividing \(w_{5}\) by \(w_{2}\): \ \(Ratio= \frac{w_{5}}{w_{2}}= \frac{1.21(1.42)^{5}}{1.21(1.42)^{2}}\)

Step 4: Simplifying the Ratio

The 1.21 term cancels out in the numerator and denominator, and the ratio simplifies to: \(Ratio= (1.42)^{5-2} = (1.42)^{3}\)

Key Concepts

Mathematical Modeling

Mathematical modeling involves creating mathematical expressions or equations to accurately represent real-world situations. In this exercise, the model given is an exponential function: \[w = 1.21(1.42)^t\]where \(w\) represents the weight of an Atlantic cod and \(t\) is the cod's age in years.
This model allows us to predict how a cod's weight increases as it ages, showing exponential growth - a concept where values increase at a rate proportional to their current value. The constants \(1.21\) and \(1.42\) were derived through empirical research and statistical analysis, representing initial conditions and growth rate, respectively.
Mathematical models are crucial in various fields, enabling scientists to predict outcomes and make informed decisions. A well-structured model simplifies complex systems into understandable equations, making problem-solving more manageable.

Ratio Calculation

Ratios help us compare two quantities showing how much of one thing there is relative to another. In our problem, we calculate the weight ratio of a 5-year-old cod to a 2-year-old cod using their weights from the exponential model. Calculating this ratio is straightforward and involves a few essential steps:

• First, find each weight by substituting the specific ages into the model equation: For 5 years: \(w_5 = 1.21(1.42)^5\)For 2 years: \(w_2 = 1.21(1.42)^2\)
• Then, compute the ratio: \[Ratio = \frac{w_5}{w_2} = \frac{1.21(1.42)^5}{1.21(1.42)^2}\]

The important observation here is how like terms \((1.21)\) cancel out, simplifying the equation significantly. The process illustrates how exponential terms behave under division; by subtracting exponents, we derive \[Ratio = (1.42)^{5-2} = (1.42)^3\] showing that the ratio depends mainly on the exponential growth factor.

Age-Weight Relationship

The relationship between age and weight in this context is expressed through an exponential function, indicating that a cod's weight increases more significantly as it grows older. This age-weight relationship is not linear, meaning that the change in weight is not constant but increases exponentially due to the factor \(1.42\) in the model.
Each passing year multiplies the previous year's weight by a factor of \(1.42\), mimicking real scenarios where organisms often grow at non-linear rates. This growth is typical in many biological organisms, where metabolic and environmental factors cause rapid changes at different life stages.
Understanding this relationship can be essential for fields like marine biology and fisheries management, allowing scientists to predict fish population dynamics and plan sustainable fishing practices. Recognizing how weight increases with age also assists in setting size limits and conservation measures, crucial for maintaining ecological balance.