Problem 16 - Algebra and Trigonometry

Question

\(7-18 \cdot\) 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

\(t = \frac{d}{55}\)

Step 1: Understand the Problem

We need to express the time taken to travel a given distance at a speed of 55 miles per hour. The goal is to express the time in terms of the given distance, denoted by variable \(d\).

Step 2: Use the Formula for Time

The basic formula to calculate time \(t\) is: \[ t = \frac{\text{distance}}{\text{speed}} \] where distance is \(d\) miles and the speed is 55 miles per hour.

Step 3: Substitute the Values

Substitute the given values into the formula: \[ t = \frac{d}{55} \]

Step 4: Write the Final Expression

The time \(t\) it takes to travel a distance \(d\) miles at a speed of 55 miles per hour is given by the expression \(t = \frac{d}{55}\). This is the required expression in terms of the variable \(d\).

Key Concepts

Distance Formula

The distance formula is a fundamental concept in algebra and physics. It establishes a relationship between the three main aspects of motion: distance, speed, and time. The formula itself can be rearranged for different calculations, but its core principle remains the same:

• Speed is how fast you are traveling (e.g. miles per hour).
• Time is how long you travel at that speed.
• Distance is how far you travel.

The standard distance formula is:\[\text{Distance} = \text{Speed} \times \text{Time}\]In our problem, we are given the speed and the distance. By rearranging the formula, we can solve for time as:\[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]This allows us to determine how long it will take for a car to travel the given distance at a specified speed. This is an essential calculation in various real-world situations, such as planning a trip or determining travel efficiency.

Speed

Speed is a measure of how quickly an object is moving. It tells us the rate at which distance is covered over a period of time. In most cases, speed is expressed in units such as miles per hour (mph) or kilometers per hour (kph).
In this exercise, the speed is a constant 55 miles per hour. This means that every hour, 55 miles are traveled. Constant speed simplifies calculations, making it easier to predict how long a journey will take over a certain distance.

• Understanding your speed is critical for maintaining schedules.
• Speed needs to be consistent with your distance units (miles with miles per hour).

By knowing the speed, we can apply the rearranged distance formula to find the time, as given in our problem, by the equation:\[\text{Time} = \frac{\text{Distance}}{55}\]This formula helps us calculate the time needed for a journey where the speed is constantly 55 mph.

Time Calculation

Time calculation is a crucial part of determining how long it will take to go from one place to another. It's all about dividing the given distance by the speed you're traveling. The equation used for time calculation is:\[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]In the given problem, this becomes:\[\text{Time} = \frac{d}{55}\]Here, "d" represents the distance in miles, while 55 is the constant speed in miles per hour.

• This equation tells us that if we know the distance and the speed, we can easily find out the travel time.
• It's important to ensure consistency in units when calculating time.
• For example, if the distance was given in kilometers, we'd need to convert it to miles first if using mph.

This principle not only helps in planning travel but also enhances our understanding of motion, allowing for more efficient scheduling and organization in everyday activities.