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Chapter 1: Problem 37

Dog Population The number of pet dogs in the United States can be represented as \(p(t)\) million where \(t\) is the number of years since \(2003 .\) (Source: Pet Food Institute) a. Write a sentence of interpretation for \(p(0)=61.5\). b. Write the function notation for the statement "There were 66.3 million pet dogs in the United States in \(2008 .\) "

Short Answer

Expert verified
a. 61.5 million pet dogs in 2003. b. \(p(5) = 66.3\).

Step by step solution

01

Understanding Function Notation

Function notation such as \(p(t)\) is used to express the number of pet dogs in millions, where \(t\) is the number of years since 2003. This means \(p(0)\) gives us the number of pet dogs in 2003.
02

Interpreting p(0) = 61.5

Since \(p(0) = 61.5\), this indicates that in 2003, there were 61.5 million pet dogs in the United States. Here, \(t = 0\) corresponds to the year 2003.
03

Determining t for 2008

The year 2008 is 5 years after 2003. Therefore, for 2008, \(t = 5\).
04

Writing Function Notation for 2008 Statement

Given "There were 66.3 million pet dogs in the United States in 2008", we can write this as \(p(5) = 66.3\), since \(t = 5\) represents the year 2008.

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Key Concepts

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

Function Notation
Function notation is an essential concept in calculus, helping us express relationships between variables directly. You can think of it as a way to replace lengthy descriptions with concise, symbolic representations. In the exercise above, the notation \( p(t) \) helps describe the number of pet dogs in the United States in millions. Here, \( t \) is the number of years since 2003.
To better understand how it works:
  • The symbol \( p \) represents the population of pet dogs.
  • \( t \) is an input that represents the passage of time, measured in years since a specific starting point (in this case, 2003).
  • \( p(t) \) gives us an output that tells us the number of dogs in any given year, calculated from 2003 onwards.
By using this notation, we simplify the process of displaying and manipulating relationships between variables, which can otherwise be complex when communicated through regular prose.
Interpretation of Functions
Interpreting functions involves extracting meaningful information from the symbolic representations used in function notation. In the exercise, interpreting \( p(0) = 61.5 \) allows us to convert abstract symbols into a real-world meaning:
Looking at \( p(0) \), it tells us:
  • The function is looked at when \( t \), the number of years since 2003, is 0.
  • The value of \( p(t) \), which accounts for the dog population, is 61.5 million.
This interpretation translates the function to mean that in the year 2003, the USA had 61.5 million pet dogs. Interpreting functions is crucial because it ties theoretical math into practical, understandable applications. It helps build a bridge between symbolic expressions and the stories they tell about the real world.
Mathematical Modeling
Mathematical modeling is the process of creating a mathematical representation of a real-world situation. In the given exercise, the function \( p(t) \) serves as a mathematical model representing the population of pet dogs in the US, dependent on the year since 2003. Creating these models involves:
  • Defining variables: \( t \) for time and \( p(t) \) for dog population.
  • Establishing relationships: The model shows how dog populations can be quantified over time.
  • Providing predictions or analyses: Using \( p(t) \), predictions about dog numbers in future or past years can be made by altering \( t \).
Mathematical modeling is powerful as it lets us simulate various scenarios and make informed decisions based on mathematical logic and historical data. It can simplify otherwise complex relationships into understandable formulas.
Time-based Variables
Time-based variables are used in mathematics to express change over time. In our exercise, \( t \) is such a variable and represents years since 2003. Here鈥檚 everything you need to know:
  • \( t = 0 \) corresponds to the baseline year of 2003, serving as a reference point.
  • For any subsequent year, you can calculate \( t \) by subtracting 2003 from the desired year.
  • Using these values in \( p(t) \) lets you observe how the dog population changes as time progresses.
This approach makes understanding trends easy because it uses a single variable to denote complex temporal changes. Using time-based variables in function notation facilitates succinct representation and manipulation of time-sensitive data, which is particularly useful in areas like economics, biology, and other sciences.

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