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) A 5 -year-old cod weighs how many times as much as a 2 -year-old cod?

Short answer

The 5-year-old cod weighs \(r\) times as much as the 2-year-old cod.

Step-by-step solution

Calculate the weight of a 2-year-old cod

Using the provided formula, the weight of a 2-year-old cod can be calculated by substituting \(t=2\) into the equation, so \(w_2=1.16(1.44)^2\). Compute the value to find the weight of a 2-year-old cod.

Calculate the weight of a 5-year-old cod

Similarly, the weight of a 5-year-old cod can be calculated by substituting \(t=5\) into the equation, so \(w_5=1.16(1.44)^5\). Compute the value to find the weight of a 5-year-old cod.

Compare the weights

The question asks for 'how many times' a 5-year-old cod weighs compared to a 2-year-old cod. This can be calculated by the ratio \(r=w_5/w_2\). Compute the ratio to find 'how many times' the 5-year-old cod weighs as much as the 2-year-old cod.

Key concepts

Algebraic Modeling

Understanding algebraic modeling involves creating mathematical equations that represent real-world situations. In our exercise, we are given a formula to model the average weight of an Atlantic cod as it ages. The equation provided, \(w = 1.16(1.44)^t\), is an example of how algebra can be used to predict changes over time, with \(w\) representing the weight of the cod in pounds and \(t\) representing the age of the cod in years.

To effectively use algebraic modeling, one needs to identify the key variables and understand how they are related. Here, the weight is determined by raising the growth factor 1.44 to the power of the age of the cod, and multiplying that by the initial weight factor, 1.16. This is a clear demonstration of how algebra can turn observations from nature into a predictable pattern described by an equation.

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value, resulting in the quantity growing by increasingly larger amounts. In the context of our problem, the weight of an Atlantic cod as it ages is represented by an exponential function.

The general form of an exponential equation is \(y = ab^x\), where \(a\) is the initial amount, \(b\) is the base or growth factor, and \(x\) is the exponent representing time or growth periods. Here, \(1.16\) is the initial factor, indicating the starting point of the weight, and \(1.44\) is the growth factor, implying that the cod's weight increases by 44% each year. This type of growth is typical in populations, finance, and, as shown in the exercise, biological growth.

Ratio Comparison

Ratio comparison is a powerful tool in mathematics for understanding the relationship between two quantities. It expresses how many times one number contains another. In our scenario, we are asked to determine how many times heavier a 5-year-old cod is compared to a 2-year-old cod.

After computing the individual weights using the exponential function given, the ratio is found by dividing the weight of the older cod by that of the younger one, \(r = w_5/w_2\). This comparison gives us a clear understanding of the relative difference in weight due to the age difference. Ratios are prevalent in various fields such as finance, engineering, and science, as they provide a simple way to express proportional relationships and make comparisons.