Problem 10 - Algebra and Trigonometry

Question

Express the given quantity in terms of the indicated variable. The time (in hours) it takes to travel a given distance at \(55 \mathrm{mi} / \mathrm{h} ; \quad d=\) given distance (in mi)

Step-by-Step Solution

Answer

The time is expressed as \(t = \frac{d}{55}\) hours.

Step 1: Understand the Problem

We need to express the time taken to travel a given distance in terms of the distance itself. We are given a speed of 55 miles per hour and a variable \(d\) representing the distance in miles.

Step 2: Use the Formula for Time

The formula to calculate time is \(t = \frac{d}{r}\), where \(t\) is time, \(d\) is distance, and \(r\) is the rate (speed). Here, \(r = 55\) mi/h.

Step 3: Substitute the Given Values

Substitute \(r = 55\) and \(d\) into the formula to get the expression for time: \(t = \frac{d}{55}\).

Step 4: Simplify the Expression

Since the speed is constant at 55 mi/h, the expression simplifies to \(t = \frac{d}{55}\) hours, where \(t\) is expressed in hours.

Key Concepts

Rate of Travel

Rate of travel is an important concept in understanding motion and how we quantify it. It tells us the speed at which an object covers a certain distance over time. In simpler terms, it describes how fast an object is moving. The standard unit of measurement for speed in the context of travel on roads or highways is usually in miles per hour (mi/h) or kilometers per hour (km/h).

• The rate of travel is consistent in a linear and straightforward path if no external factors, like traffic, affect it.
• A higher speed means a shorter amount of time is needed to cover the same distance.

In our example, the rate of travel is given as 55 mi/h. This implies that for every hour, the object moves 55 miles. Knowing the rate of travel helps us predict how long a journey will take if we know the distance.

Time Calculation

Time calculation is crucial when planning travel to ensure punctuality and resource management. The time taken for a journey depends on both the distance to be traveled and the speed of travel. To calculate the time, we use the following formula:
\[ t = \frac{d}{r} \]Where:

• \( t \) is the time in hours.
• \( d \) is the distance in miles.
• \( r \) is the speed in miles per hour.

This formula derives from rearranging the basic distance formula. Since time is the duration taken to cover a distance at a certain rate, it inversely depends on speed; a higher speed requires less time, and vice versa. Thus, by substituting the given distance and speed into this formula, one can determine the exact time required for the trip.

Distance Formula

The distance formula helps us understand the relationship between speed, time, and distance. It is expressed as:
\[d = r \times t\]Where:

• \( d \) is the distance.
• \( r \) is the rate of travel or speed.
• \( t \) is the time.

This formula is foundational in physics and mathematics to compute the distance traveled when speed and time are known. It is equally useful for rearranging to solve for any variable in travel-related problems.
In our exercise, we started with the fundamental relationship of time and then used this to create a modified version by solving for time instead of distance. By organizing the formula like this, it allows easy manipulation for various travel scenarios, like determining stopping points or durations. Understanding that distance is simply a product of time and speed can simplify problem-solving in motion-related queries.