Problem 47 - College Algebra with Corequisite Support

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

Answer

The car takes 3.6 hours to travel 180 miles.

Step 1: Understand the Relationship

The function given is \(d(t) = 50t\), which indicates that the distance \(d\) is dependent on time \(t\). This tells us at any time \(t\), the distance traveled is 50 times \(t\).

Step 2: Set up the Inversion

To find the inverse, we need an expression for time \(t\) in terms of distance \(d\). Thus, we want to isolate \(t\) in the equation \(d(t) = 50t\).

Step 3: Solve for Time (t)

To isolate \(t\), divide both sides of the equation by 50: \(t(d) = \frac{d}{50}\). Now, the function \(t(d)\) gives time as a function of distance.

Step 4: Calculate Specific Value

Use the inverse function \(t(d) = \frac{d}{50}\) to calculate \(t(180)\). Substitute \(d = 180\) into the expression: \[t(180) = \frac{180}{50} = 3.6\].

Step 5: Interpret the Result

The result \(t(180) = 3.6\) means that when the car travels 180 miles, the time taken is 3.6 hours.

Key Concepts

Function of Time

Understanding functions can help us relate different concepts, like time and distance, in mathematical problems. When we talk about the "function of time," we're examining how time affects another variable—in this case, distance. In the given problem, distance is expressed as a function of time, which means that the distance traveled by the car depends on how long it has been traveling. A function of time can be thought of as a rule that assigns a unique output (distance) for each input (time).
\(d(t) = 50t\) is the particular function representing this scenario, stating that for every hour the car travels, it covers 50 miles. Thus, if you know the time, you can calculate the distance easily by plugging the time into the function. This straightforward relationship highlights how measurable quantities can change as time progresses.

Distance and Speed Relationship

The relationship between distance and speed is one of the core concepts taught in physics and mathematics. Distance being a function of speed and time can be captured by the simple formula:

• Distance = Speed \(\times\) Time

In this problem, the car travels at a constant speed of 50 miles per hour. This speed remains the same regardless of how long the car travels, which simplifies our calculations.As speed is constant, the relationship is linear and straightforward. Any change in time leads to a proportional change in distance. For example, doubling the time results in doubling the distance traveled, assuming the speed stays the same. It is important to note that this formula gives a direct method to calculate one of the variables (distance, speed, time) if the other two are known. This is why understanding the distance-speed relationship is crucial in both basic and advanced mathematical contexts.

Mathematical Interpretation

Mathematical interpretation involves analyzing functions and their inverses to understand what they represent in real-world terms. In the provided exercise, the inverse function \(t(d) = \frac{d}{50}\) represents time as a function of distance, which flips the original function \(d(t) = 50t\). This means instead of finding out how far the car will travel in a given time, you can determine how long it will take to travel a certain distance.
For example, when calculating \(t(180)\), you substitute the distance (180 miles) into the inverse function to find the time, resulting in 3.6 hours. This interpretation helps us understand how functions and their inverses provide different perspectives on the same problem. By expressing time as a function of distance, we can make predictions, plan schedules, or calculate durations for trips effectively. So, inverse functions give us valuable tools to reverse our calculations and obtain insights from different angles.