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Chapter 3: Problem 41

Solve the given problems. A truck travels \(300 \mathrm{km}\) in \(t-3 \mathrm{h}\). Express the average speed \(s\) of the truck as a function of \(t .\) What are the domain and range of \(s=f(t) ?\)

Short Answer

Expert verified
The function is \(s(t) = \frac{300}{t-3}\). Domain: \(t \in \mathbb{R}, t \neq 3\). Range: \(s \in \mathbb{R}, s \neq 0\).

Step by step solution

01

Understand the Problem

We need to express the average speed of a truck as a function of time and find the domain and range of this function. The truck travels 300 km in a time duration of \(t-3\) hours.
02

Write the Formula for Average Speed

The formula for average speed \(s\) is given by \(s = \frac{\text{Total Distance}}{\text{Total Time}}\). In this case, the total distance is 300 km, and the total time is \(t-3\) hours.
03

Express Speed as a Function

Substitute the values into the formula: \[ s = \frac{300}{t-3} \] This equation expresses the average speed as a function of \(t\), denoted as \(s = f(t)\).
04

Determine the Domain of the Function

The domain of a function is the set of all possible input values (in this case, \(t\)) that the function can accept. Since division by zero is undefined, \(t-3\) cannot be zero. Therefore, \(t eq 3\). The domain is all real numbers except 3: \(t \in \mathbb{R}, t eq 3\).
05

Determine the Range of the Function

The range is all possible output values of the function. Since \(s = \frac{300}{t-3}\), \(s\) can take any non-zero real number value (positive if \(t > 3\), negative if \(t < 3\)). Thus, the range is all real numbers excluding zero: \(s \in \mathbb{R}, s eq 0\).

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Key Concepts

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

Domain and Range
In mathematics, the domain and range of a function are essential concepts that describe where a function can take, or output, its values. The domain is the complete set of possible values of the independent variable, which is often represented by the letter \( t \) in this context. For our problem, we are asked to find the domain of the function \( s = f(t) = \frac{300}{t-3} \).

Let's break it down:
  • The denominator \( t-3 \) cannot equal zero, as division by zero is undefined. Therefore, \( t eq 3 \).
  • Thus, the domain is all real numbers except when \( t = 3 \). We express this as \( t \in \mathbb{R}, t eq 3 \).
The range, on the other hand, represents all possible output values. Since \( s = \frac{300}{t-3} \), \( s \) can take any real value except zero:
  • If \( t > 3 \), \( s \) is positive.
  • If \( t < 3 \), \( s \) is negative.
Consequently, the range is all real numbers excluding zero: \( s \in \mathbb{R}, s eq 0 \).
Function of Time
Expressing speed as a function of time is crucial in understanding how speed changes based on the time variable. In our problem, the truck's average speed is given as a function of time, \( s = f(t) \), where:\[s = \frac{300}{t-3}\]This equation indicates that:
  • The speed \( s \) depends on the time \( t \) measured from 3 hours earlier.
  • Speed increases as the denominator \( t-3 \) increases, meaning time passes beyond 3 hours.
Understanding this function aids in visualizing how speed is not static. Instead, it varies dynamically based on the time elapsed since the starting point. Another key takeaway is recognizing how functions of time help predict behavior and can be graphically shown to give insights into trends and patterns. Remember, time functions can either increase, decrease, or shift symmetrically based on its defining equation.
Rational Functions
A rational function is a type of function that can be expressed as the quotient of two polynomials. Our problem's average speed function \( s = \frac{300}{t-3} \) is a classic example of a rational function:
  • The numerator is the constant polynomial \( 300 \).
  • The denominator is a linear polynomial \( t-3 \).
Rational functions like these have distinct characteristics:
  • They are undefined wherever the denominator equals zero. For our function, this happens at \( t = 3 \).
  • They can create vertical asymptotes at these points, where the function values can become unbounded. This unbounded behavior is a critical point in understanding the nature of rational functions.
  • The graphs of rational functions can also exhibit horizontal asymptotes, indicating end behavior where the function approaches a line but never actually meets it.
Rational functions are essential in predicting real-world scenarios where ratios or rates are involved, helping analyze phenomena such as speed, growth, and decay across various fields.

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