Problem 47
Question
A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, \(t,\) in hours given by \(d(t)=50 t\). Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function \(t(d)\). Find \(t(180)\) and interpret its meaning.
Step-by-Step Solution
Verified Answer
\(t(180) = 3.6\); it takes 3.6 hours to travel 180 miles.
1Step 1: Understand the Problem
We have a function \(d(t) = 50t\) which gives us the distance traveled by the car in miles as a function of time \(t\) in hours. We need to find the inverse function \(t(d)\), which will express time \(t\) in terms of the distance \(d\) traveled.
2Step 2: Set Up the Equation for the Inverse Function
Start with the original function: \(d = 50t\). To find the inverse, we need to express \(t\) in terms of \(d\). Thus, we will be solving for \(t\) in the equation \(d = 50t\).
3Step 3: Solve for Time in Terms of Distance
The equation \(d = 50t\) can be rearranged to solve for \(t\). Divide both sides by 50 to get \(t = \frac{d}{50}\). This gives us the inverse function \(t(d) = \frac{d}{50}\).
4Step 4: Evaluate the Inverse Function for a Specific Distance
We have \(t(d) = \frac{d}{50}\). To find \(t(180)\), substitute 180 for \(d\): \(t(180) = \frac{180}{50} = 3.6\).
5Step 5: Interpret the Meaning of \(t(180)\)
The value \(t(180) = 3.6\) indicates that it takes 3.6 hours for the car to travel 180 miles at the constant speed of 50 miles per hour.
Key Concepts
Understanding Constant SpeedExploring Distance-Time RelationshipFunction Notation and Its Importance
Understanding Constant Speed
When an object moves at a constant speed, it means it covers equal distances in equal intervals of time.
This is a simple but important concept in physics and mathematics. Imagine a car moving steadily without speeding up or slowing down.
For example, if a car travels at a constant speed of 50 miles per hour, it will cover:
In our exercise, this constant speed provides a direct relationship between distance and time, encapsulated by the function \(d(t)=50t\).
This means, essentially, that for every hour \(t\) traveled, 50 miles are covered, assuming the speed doesn't change.
This is a simple but important concept in physics and mathematics. Imagine a car moving steadily without speeding up or slowing down.
For example, if a car travels at a constant speed of 50 miles per hour, it will cover:
- 50 miles in one hour
- 100 miles in two hours
- 150 miles in three hours
In our exercise, this constant speed provides a direct relationship between distance and time, encapsulated by the function \(d(t)=50t\).
This means, essentially, that for every hour \(t\) traveled, 50 miles are covered, assuming the speed doesn't change.
Exploring Distance-Time Relationship
The distance-time relationship is key when quantifying motion or travel.
Bearing constant speed in mind, distance traveled is directly proportional to the time of travel.
In other words, if you know how much time has passed, you can calculate how far something has traveled if it moves at a constant pace.
Mathematically, we express this relationship with a function: \(d(t) = 50t\).
This particular function shows how distance \(d\) changes over time \(t\) for an object moving at 50 miles per hour.
But what if you wish to know how long it took to cover a certain distance?
This requires calculating the inverse of the distance-time function.Finding the inverse function \(t(d)\) means rearranging the relationship to solve for time \(t\) as a function of distance \(d\).
This inverse function is important because it allows us to deduce the time needed for any given distance at the set speed.
For instance, if our destination is 180 miles away, and we're traveling at 50 mph, by substituting the distance into \(t(d) = \frac{d}{50}\), we determine it will take 3.6 hours to reach there.
Bearing constant speed in mind, distance traveled is directly proportional to the time of travel.
In other words, if you know how much time has passed, you can calculate how far something has traveled if it moves at a constant pace.
Mathematically, we express this relationship with a function: \(d(t) = 50t\).
This particular function shows how distance \(d\) changes over time \(t\) for an object moving at 50 miles per hour.
But what if you wish to know how long it took to cover a certain distance?
This requires calculating the inverse of the distance-time function.Finding the inverse function \(t(d)\) means rearranging the relationship to solve for time \(t\) as a function of distance \(d\).
This inverse function is important because it allows us to deduce the time needed for any given distance at the set speed.
For instance, if our destination is 180 miles away, and we're traveling at 50 mph, by substituting the distance into \(t(d) = \frac{d}{50}\), we determine it will take 3.6 hours to reach there.
Function Notation and Its Importance
Function notation is a fundamental way to describe mathematical relationships. It shows how one quantity depends on another.
For our problem, function notation is used to express how distance depends on time and also how time can depend on distance in the case of the inverse function.
For instance, the notation \(d(t)\) denotes a function where distance \(d\) is dependent on time \(t\).
Similarly, \(t(d)\) is the inverse function notation, expressing how time depends on distance.
This notation helps in systematically calculating and interpreting different mathematical relationships easily.
By using these notations, you can plug numbers into the function to quickly find results.
This becomes particularly beneficial when examining ways in which quantities influence each other, either directly or inversely, within real-world contexts like travel and physics.
For our problem, function notation is used to express how distance depends on time and also how time can depend on distance in the case of the inverse function.
For instance, the notation \(d(t)\) denotes a function where distance \(d\) is dependent on time \(t\).
- It's like saying "the output \(d\) varies with input \(t\) over the period of travel".
Similarly, \(t(d)\) is the inverse function notation, expressing how time depends on distance.
This notation helps in systematically calculating and interpreting different mathematical relationships easily.
By using these notations, you can plug numbers into the function to quickly find results.
This becomes particularly beneficial when examining ways in which quantities influence each other, either directly or inversely, within real-world contexts like travel and physics.
Other exercises in this chapter
Problem 46
The circumference \(C\) of a circle is a function of its radius given by \(C(r)=2 \pi r\). Express the radius of a circle as a function of its circumference. Ca
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For the following exercises, given each function \(f,\) evaluate \(f(-3), f(-2), f(-1),\) and \(f(0)\) $$ f(x)=\left\\{\begin{array}{ccc}{x+1} & {\text { if }}
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For the following exercises, determine whether the function is odd, even, or neither. $$ f(x)=3 x^{4} $$
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Determine whether the function is odd, even, or neither. $$f(x)=3 x^{4}$$
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