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Chapter 5: Problem 2

A class of models for population growth rates in marine fisheries assumes that the harvest from fishing is proportional to the population size. One such model uses a quadratic function: $$ G=0.3 n-0.2 n^{2} . $$ Here \(G\) is the growth rate of the population, in millions of tons of fish per year, and \(n\) is the population size, in millions of tons of fish. a. Make a graph of \(G\) versus \(n\). Include values of \(n\) up to \(1.7\) million tons. b. Calculate \(G(1.62)\) and explain what your answer means in practical terms. c. At what population size is the growth rate the largest?

Short Answer

Expert verified
Graph shows growth rate vs. population. At \( n = 1.62 \), growth is slightly negative. Maximum growth occurs at \( n = 0.75 \) million tons.

Step by step solution

01

Understanding the Function

The function given is a quadratic equation: \( G = 0.3n - 0.2n^2 \). Here, \( G \) represents the growth rate of the population in millions of tons per year, and \( n \) is the population size in millions of tons of fish.
02

Graphing G vs n

To graph \( G \) versus \( n \), evaluate the function for various values of \( n \) from \( 0 \) to \( 1.7 \). Calculate a few specific points such as \( n = 0, 0.5, 1.0, 1.5, \) and \( 1.7 \). Plot these values on a graph, placing \( n \) on the x-axis and \( G \) on the y-axis. Connect the points with a smooth curve which should resemble an inverted parabola, indicating the growth rate increases and then decreases as population size increases.
03

Calculating G(1.62)

Substitute \( n = 1.62 \) into the equation: \( G(1.62) = 0.3(1.62) - 0.2(1.62)^2 \). This results in \( G(1.62) = 0.486 - 0.52488 \), which simplifies to \( G(1.62) = -0.03888 \). This means the population's growth rate is slightly negative at this population size, indicating a small decline in population at this size.
04

Finding the Maximum Growth Rate

The function \( G = 0.3n - 0.2n^2 \) is quadratic and opens downward (since the coefficient of \( n^2 \) is negative). To find the maximum growth rate, identify the vertex of this parabola. For a quadratic \( an^2 + bn + c \), the vertex occurs at \( n = -\frac{b}{2a} \). Here, \( a = -0.2 \) and \( b = 0.3 \), so \( n_{max} = -\frac{0.3}{2(-0.2)} = 0.75 \). Thus, the growth rate is largest when the population size is 0.75 million tons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
Quadratic functions are expressions where the highest degree is 2. They generally have the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In this context, the quadratic function \( G = 0.3n - 0.2n^2 \) models the growth rate of a fish population based on its size. Here, \( G \) represents the growth rate and \( n \) is the population size, both measured in millions of tons.
This particular function is important because it describes how the growth rate changes as the population grows. When analyzing population dynamics, understanding the behavior of quadratic functions can help predict whether a population is likely to increase or decrease. Since the graph of this quadratic function opens downward (due to the negative coefficient of \( n^2 \)), it indicates a maximum point beyond which further increase in population size would lead to a decrease in growth rate.
Graphing Techniques
Graphing techniques are essential to visualize quadratic functions and better understand their behavior. To graph the function \( G = 0.3n - 0.2n^2 \), you need to evaluate the function at various points between \( n = 0 \) and \( n = 1.7 \). This involves substituting values such as 0, 0.5, 1.0, 1.5, and 1.7 into the function to find corresponding \( G \) values.
The resulting graph is a parabola opening downward. This shape shows where the growth rate increases and then starts to decrease as the population size gets larger. By connecting these calculated points with a smooth curve, you can clearly observe the basic characteristics of the function, such as the decrease in growth rate past a certain population size. Graphs provide a visual tool that can simplify the complexity of algebraic expressions, making patterns and trends more apparent.
Vertex of a Parabola
The vertex of a parabola is a key point that reveals the maximum or minimum value of a quadratic function. For a downward-opening parabola like \( G = 0.3n - 0.2n^2 \), the vertex represents the maximum growth rate.
The formula \( n = -\frac{b}{2a} \) is used to find the vertex. In our case, \( a = -0.2 \) and \( b = 0.3 \). This gives \( n = -\frac{0.3}{2(-0.2)} = 0.75 \), meaning the maximum growth rate occurs when the fish population is 0.75 million tons. This vertex concept helps us understand at what population size resource constraints or other factors start to limit growth. In population models, recognizing this point is crucial for managing resources sustainably, ensuring populations aren't pushed to sizes that lead to negative growth rates.

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Most popular questions from this chapter

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