Question

Write an algebraic expression for the verbal expression. The distance traveled in \(t\) hours by a car traveling at 50 miles per hour

Short answer

\(d = 50t\)

Step-by-step solution

Understand the problem

Firstly, a rate of 50 miles per hour means that the car will travel 50 miles every hour. The time the car has been traveling for is represented by \(t\) hours.

Translate to algebraic expression

In defining the distance, \(d\), travelled by a car, the formula is \(d = rt\), where r is rate of travel and t is time. Here, the rate is 50 miles per hour and time is \(t\) hours. Therefore, replacing \(r\) with 50 and \(t\) with \(t\) in the formula gives \(d = 50t\)

Key concepts

Distance Formula

Understanding the distance formula is crucial when solving problems related to movement over time. The formula forms a fundamental relationship between distance, rate, and time, often written as \( d = rt \). In this equation, \(d\) represents the total distance traveled, while \(r\) stands for the rate of travel (also could be the speed at which an object is moving), and \(t\) denotes the time spent traveling. The beauty of the distance formula lies in its simplicity; it allows for easy calculation of any of the three variables as long as the other two are known.

For instance, in the given exercise, if a car is traveling at a constant speed, the distance the car travels over a period of time can be calculated by multiplying the rate (speed) by the time. It eliminates the need for complicated computations and provides a clear and direct way to determine how far the car has gone during a particular timeframe. This formula is not just for cars; it can be applied to any object moving at a steady rate, making it a versatile tool in algebra.

Rate of Travel

The rate of travel, often measured in units such as miles per hour or kilometers per hour, indicates the speed at which an object is traveling over a certain period. It is a comparison of the distance traveled to the time taken. When solving algebraic expressions involving movement, understanding the rate of travel is essential.

As seen in our example, a car traveling at a rate of 50 miles per hour is moving fast enough to cover 50 miles in a single hour. This information allows us to determine the distance covered over any period, simply by multiplying the rate by the time, as indicated in the distance formula. It's important to maintain consistent units when calculating the rate of travel to ensure accurate results. The rate is a key variable in the distance formula and serves as the connector between time and distance.

Practical Application of Rate

In real-life scenarios, such as planning a trip, the rate of travel helps in estimating arrival times and distances that can be covered in a day.

Variable Representation

Algebra is all about using symbols, or variables, to stand in for unknown values. Variable representation is a way to summarize and solve problems without having exact numbers for every term. In the context of our exercise, the variable \(t\) is used to represent the time in hours, which is not specified and can be any number.

By using variables, algebra allows us to create a formula that can solve not just one problem but many similar ones, regardless of the specific values. Once the value of a variable is known, it can be substituted into the equation to find the answer to a particular scenario. For example, with the algebraic expression \(d = 50t\), if we know that \(t = 3\) hours, we can easily find that \(d = 150\) miles. Variable representation is a powerful tool that makes algebra a universally applicable branch of mathematics.