Question

The table shows the average weight (in kilograms) of an Atlantic cod that is \(x\) years old from the Gulf of Maine. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Age, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Weight, } \boldsymbol{y} & 0.751 & 1.079 & 1.702 & 2.198 & 3.438 \\ \hline \end{array} $$ a. Show that an exponential model fits the data. Then find an exponential model for the data. b. By what percent does the weight of an Atlantic cod increase each year in this period of time? Explain.

Short answer

The exponential model fitting the given data is given by \(P = 0.751 * e^{kt}\), where \(k\) is approximately the yearly percent increase in decimal form. Therefore, the actual percentage increase is \(k * 100\)% per year.

Step-by-step solution

Identify the Given Data

The data given shows the weight of an Atlantic cod at different ages. The age is denoted by \(x\) and the weight by \(y\). The weight appears to be increasing with the age, hinting at a possible exponential growth model.

Find the Exponential Model

An exponential growth can be represented by the formula \(P = P0 * e^{kt}\) where \(P0\) is the starting quantity (the weight when \(x=1\)), \(k\) is the rate of growth, and \(t\) is time. Judging by the data, specific values of \(x\) and \(y\) can be plugged into the formula to find the growth rate \(k\).

Calculate the Growth Rate

Choosing the weights for age \(x=1\) and \(x=2\), and solving for \(k\) gives: \(1.079 = 0.751 * e^{k(2-1)} \Rightarrow k = ln(1.079/0.751)\), where \(ln()\) is the natural logarithm. Using the calculated value of \(k\), the exponential growth model can be written as \(P = 0.751 * e^{kt}\).

Calculate the Yearly Percent Increase

The value of \(k\) can be interpreted as the yearly percent increase in the weight of the Atlantic cod in decimal form. To get the percentage form, multiply \(k\) by 100.

Explanation of the Yearly Percent Increase

The yearly percent increase is the proportion that the weight of the Atlantic cod increases each year, meaning with each subsequent year, the weight increases by an amount proportional to the current weight.

Key concepts

Exponential Model

An exponential model is a mathematical representation where the rate of change of a quantity is proportional to its current value. This is common in scenarios involving growth, like population or weight growth over time.

• In the context of the Atlantic cod weight as it ages, we notice that the weight doesn't just increase by a constant amount each year but grows more significantly as time passes.
• This suggests an exponential growth, where the model captures how the cod's weight grows by a multiplier each year.
• The general form for this is given by the equation \(P = P_0 \times e^{kt}\), where \(P_0\) is the initial weight and \(e\) is the base of the natural logarithm.

In this exercise, we determined that checking how the weight relative to time follows this general form is crucial to modeling the exponential growth.

Growth Rate

The growth rate is an essential part of any exponential model. It indicates how rapidly a quantity grows over time. In the formula \(P = P_0 \times e^{kt}\), the rate \(k\) is what we're interested in.

• This rate is crucial because it tells us how fast our quantity, in this case, the weight of the cod, is increasing relative to its size.
• By solving with specific examples from the data, like \(x=1\) and \(x=2\), we calculate \(k\) using logarithmic methods (\(k = \ln(1.079/0.751)\)).
• The growth rate \(k\) informs us about the natural tendency of the cod's weight to increase over time.

Remember, a higher growth rate \(k\) means faster exponential growth.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a key mathematical function in dealing with exponential growth. It helps simplify the manipulation of equations involving exponentials.

• \(\ln\) is the inverse of the exponential function, \(e^x\), allowing us to unravel the exponential equations and solve for \(k\) as seen in the formula \(k = \ln(\text{ratio})\).
• \(\ln(e)\) equals 1, which is a crucial identity when manipulating exponential functions.
• The capability to turn multiplicative processes into additive ones makes \(\ln\) invaluable for calculating continuous compounding, like in this cod's weight problem.

Using \(\ln\) in the exercise allowed us to extract the growth rate \(k\) efficiently.

Percent Increase

The percent increase reflects how much a quantity grows in percent terms yearly. It translates the growth rate into a more digestible number.

• In our exponential model, \(k\) is in decimal or fraction form, which may not be immediately clear how large it is in terms of percentages.
• To convert \(k\) into percent, multiply it by 100, giving a clearer measure of the Atlantic cod's annual weight increase.
• This percentage shows the relative change compared to the initial weight and helps quantify yearly growth patterns.

For example, if \(k\) is 0.32, then the percent increase is 32%, stating the weight grows 32% larger each year.

Mathematical Modeling

Mathematical modeling is a technique for representing real-world scenarios using mathematical expressions and equations. It's very powerful for predicting and understanding complex systems.

• In this exercise, we're using a model to describe how the average weight of an Atlantic cod changes as it grows older.
• Mathematical modeling helps identify trends and make future predictions about a system—in this case, helping fisheries and scientists understand cod growth.
• Models can be adjusted with new data or refined if the initial assumptions were off, making them versatile tools for ongoing assessment.

Such modeling is fundamental in fields like biology, economics, and any discipline where growth prediction is essential.