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Chapter 2: Problem 6

The amount \(C\) of food consumed in a day by a sheep is a function of the amount \(V\) of vegetation available, and a model is $$ C=\frac{3 V}{50+V} $$ Here \(C\) is measured in pounds and \(V\) in pounds per acre. This relationship is called the functional response. a. Make a graph of \(C\) versus \(V\). Include vegetation levels up to 1000 pounds per acre. b. Calculate \(C(300)\) and explain in practical terms what your answer means. c. Is the graph concave up or concave down? Explain in practical terms what this means. d. From the graph it should be apparent that there is a limit to the amount of food consumed as more and more vegetation is available. Find this limiting value of \(C\).

Short Answer

Expert verified
a. Graph: Plot concave down. b. C(300) 鈮� 2.57 pounds. c. Graph is concave down, consumption levels off. d. Limiting value is 3 pounds.

Step by step solution

01

Understanding the Function

The given function is \[ C(V) = \frac{3V}{50+V} \]This represents how much food, in pounds, a sheep will consume based on the pounds of vegetation per acre available, \(V\). Our task involves analyzing this function for different parts of the exercise, including plotting, calculating, and determining its behavior as \(V\) increases.
02

Graphing the Function

To graph \(C(V) = \frac{3V}{50+V}\) versus \(V\), we plot the function for values of \(V\) ranging from 0 to 1000. Start by creating a table of values for specific \(V\) values and compute corresponding \(C(V)\). Example values might include \(V = 0, 50, 100, 500,\) and \(1000\). Analyze and plot these points using graphing software or graph paper and sketch the curve smoothly passing through these points.
03

Calculating C(300)

To find \(C(300)\), substitute \(V = 300\) into the formula:\[ C(300) = \frac{3 \times 300}{50 + 300} = \frac{900}{350} = \frac{18}{7} \approx 2.57 \]This result means that when there are 300 pounds of vegetation per acre, a sheep will consume approximately 2.57 pounds of food per day.
04

Analyzing Concavity

To determine if the graph is concave up or concave down, we need the second derivative of \(C(V)\). Without calculating, observe that this type of equation is similar to a "saturation" curve or Michaelis-Menten form, which is typically concave down, indicating initial rapid increases in consumption followed by leveling off. Practically, this means sheep eat rapidly at first with more vegetation, but the additional vegetation doesn't significantly increase the amount consumed beyond a point.
05

Finding the Limiting Value

To find the limit of \(C(V)\) as \(V\) approaches infinity, we compute:\[ \lim_{V \to \infty} \frac{3V}{50+V} \]As \(V\) increases, the \(50+V\) approaches \(V\), so:\[ \lim_{V \to \infty} \frac{3V}{V} = 3 \]This means that no matter how much vegetation is present, the sheep's consumption will not exceed 3 pounds per day. This is the maximum capacity of consumption per day due to biological constraints.

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Key Concepts

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

Concave Down Nature of the Graph
In mathematics, when describing whether a curve is concave up or concave down, we look at its bending direction. For our function \[ C(V) = \frac{3V}{50+V} \], we focus on its concavity.
The "concave down" indicates that as the curve rises, it does so in a way that becomes less steep as you move from left to right.
Think of it as an upside-down bowl shape.
The graph of the function rapidly increases at first when there鈥檚 little vegetation available. But as more vegetation is added, the increase in consumption tapers off.
This is typical of "saturation" models or biological responses where there鈥檚 an initial burst of activity that slows gradually due to limitations, in this case, the sheep鈥檚 eating capacity.
Another way to view concavity is through the second derivative. If the second derivative is negative, the graph is concave down.
  • Initial rapid consumption with small amounts of vegetation.
  • Consumption growth slows down despite more vegetation.
  • This reflects biological limits in eating capacity.
Limiting Value of the Functional Response
When exploring functions, especially in biology or economics, understanding the "limiting value" is crucial. Here, our model represents a biological constraint or maximum:\( \lim_{V \to \infty} \frac{3V}{50+V} \).
As the vegetation \( V \) increases, the denominator \( 50+V \) approaches \( V \), simplifying the expression to approach 3 as \( V \) becomes very large.
This tells us that the sheep鈥檚 maximum food consumption rate is 3 pounds per day.
It doesn鈥檛 matter how much extra vegetation you provide, beyond a point, the sheep's intake will not increase.
This limiting behavior can be thought of as the biological or physical limit of consumption for the sheep, and serves as a crucial insight when modeling natural phenomena with mathematical functions.
  • Represents the "cap" or maximum capacity for consumption.
  • Shows limitations in biological systems.
  • Important for distinguishing realistic models versus unbounded growth.
Understanding the Graphing of Functions
Graphing functions is key to visualizing and understanding their behavior. For the function \( C(V) = \frac{3V}{50+V} \), graphing reveals how consumption changes as vegetation increases.
Start by selecting a range of \( V \) values, such as 0 to 1000 pounds per acre.
Using these values, calculate corresponding \( C(V) \) and plot these points on a graph.
This can demonstrate how quickly consumption rises initially and where it levels off.
The curve you鈥檒l observe is smooth and doesn't have sharp turns or zigzags, reflecting the model鈥檚 realistic biological response.
Graphing tools or graph paper can help in accurately plotting these values.
  • Visualize the initial rapid increase in consumption.
  • Note where consumption plateaus, indicating the limiting value.
  • Use tools to ease plotting for accurate results.

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Most popular questions from this chapter

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