Problem 54 - 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) Find the ratio of the weight of a 5 -year-old cod to the weight of a 2 -year- old cod. Express this ratio as a power of 1.44

Step-by-Step Solution

Answer

The ratio of the weight of a 5-year-old cod to the weight of a 2-year-old cod is \(1.44^{3}\).

Step 1: Identify the given information

The ages of the Atlantic cods are given as 5 years and 2 years. Using the provided model \(w=1.16(1.44)^{t}\), the weights of Atlantic cods of these ages respectively can be computed. Let \(w_{5}\) be the weight of the 5-year-old cod and \(w_{2}\) be the weight of the 2-year-old cod.

Step 2: Calculate the weight of a 5-year-old cod using the model

Substitute \(t = 5\) into the model to get \(w_{5} = 1.16(1.44)^{5}\).

Step 3: Calculate the weight of a 2-year-old cod using the model

Substitute \(t = 2\) into the model to get \(w_{2} = 1.16(1.44)^{2}\).

Step 4: Find the ratio of the weights

The ratio of the weights of the 5-year-old cod to the 2-year-old cod can be found by dividing \(w_{5}\) by \(w_{2}\). So, \(\frac{w_{5}}{w_{2}} = \frac{1.16(1.44)^{5}}{1.16(1.44)^{2}}\).

Step 5: Simplify the ratio

Simplify the ratio by cancelling out the common factors in the numerator and the denominator. The constant factor 1.16 cancels out and the ratio simplifies to \(1.44^{(5-2)} = 1.44^{3}\).

Key Concepts

Algebraic Modeling

Algebraic modeling is a method used to describe real-life situations using mathematical equations or expressions. It helps in predicting outcomes by representing variables in a clear and structured way. In our case, the average weight of an Atlantic cod as it ages is modeled using the equation \(w = 1.16(1.44)^t\). This formula indicates how the weight changes over time, with \(t\) representing the cod's age in years.

When we say that the weight can be modeled by this equation, it means that, regardless of the complexity of the biological processes involved, this mathematical model provides an approximate prediction of a cod's weight. Such equations are incredibly useful in biology and many other fields to predict trends and make informed decisions.

Key components of algebraic modeling include:

• Variables: \(w\) and \(t\) where \(w\) is a function of \(t\).
• Constants: The numbers in the equation such as 1.16 and 1.44
• Operations: The operations help demonstrate the relationship between variables and constants, such as exponentiation indicated by \(1.44^t\).

This simple model takes a lot of real-world complexity and condenses it into a manageable form that can be used for calculations and predictions.

Weight Ratio

Understanding weight ratios helps us compare different values to see their relationship to one another. In this exercise, we compare the weight of a 5-year-old cod to that of a 2-year-old cod.

To find this, we use their respective weights using the algebraic model: \( w_5 = 1.16(1.44)^5 \) and \( w_2 = 1.16(1.44)^2 \). The ratio of these weights is calculated by dividing the weight of the 5-year-old by that of the 2-year-old:

\[ \frac{w_{5}}{w_{2}} = \frac{1.16(1.44)^{5}}{1.16(1.44)^{2}} \]

When we look at ratios, any common factors in the equation, like 1.16 in this case, can be cancelled out. This leaves us with the power term, which simplifies to \( 1.44^{5-2} \) or \( 1.44^3 \).

By expressing this ratio as a power of 1.44, it becomes clear how the weight changes proportionately over 3 years. Such ratios are extremely valuable in making comparisons not just in biology, but also in finance, cooking, and everyday decision-making situations.

Simplifying Expressions

Simplifying expressions is a key mathematical skill involving the process of making a mathematical expression more concise without changing its value. For our exercise, it's crucial to simplify to better understand and work with equations.

In the step-by-step solution, the simplification process involves:

• Identifying common factors in the numerator and denominator of a fraction. Here, both the numerator and denominator had a common factor of 1.16 in the weight ratio equation.
• Performing subtraction on exponents when like bases exist. This is seen when \( (1.44)^5 \) is reduced with \( (1.44)^2 \) resulting in \( 1.44^{5-2} \), simplifying to \( 1.44^3 \).

Why is this important? By simplifyingexpressions, you achieve a clearer understanding and more efficient calculation process. Simplified expressions often reveal insights into the behavior of the system being modelled, making it easier to draw conclusions and make informed decisions.