Question

Write an algebraic expression for the verbal expression. The travel time for a plane that is traveling at a rate of \(r\) miles per hour for 200 miles

Short answer

The algebraic expression is \( T = \frac{200}{r} \) hours

Step-by-step solution

Identify the given variables

The speed of the plane is given as \(r\) miles per hour and the distance it is going to cover is 200 miles.

Apply the speed, distance, time formula

We want to find the time the plane will take, which we can find by the formula time = distance/speed. In this case that will be time = 200 miles / \(r\) miles per hour

Write down the algebraic expression

The algebraic expression representing the travel time for a plane that is traveling at a rate of \(r\) miles per hour for 200 miles becomes: \( T = \frac{200}{r} \) hours

Key concepts

Speed Distance Time Formula

The speed, distance, time formula is an essential tool in physics and mathematics for solving problems related to motion. It's used to calculate one of the three variables when the other two are known. The formula can be expressed as:

Time = Distance / Speed

Using this formula allows us to understand the relationship between how fast something is moving (speed), how much ground it has covered (distance), and how long it has been moving (time).

Real-World Applications

Transportation is a prime example where this formula is frequently applied. For instance, calculating trip durations or determining the best speed to travel to reach a destination on time. Let's imagine a situation where a car travels 150 miles at a constant speed of 50 miles per hour. Using the formula, we calculate the time taken for the journey as 3 hours (Time = 150 / 50).
In the educational context, problems involving this formula help students understand the interplay of these three variables in a tangible way. Through practice, they learn to solve for any one of the variables when provided with the other two, honing their problem-solving and critical-thinking skills.

Variables in Algebra

Variables are the building blocks of algebra. They are symbols used to represent numbers in equations and expressions, allowing us to generalize problems and work with unknown values. The most common variable names are simple letters like 'x', 'y', and 'z', but they can be any symbol, including Greek letters or even words.

Constant vs. Variable

A variable can change and take on different values within the context of a problem. In contrast, a constant is a fixed value that does not change. For example, in the formula for the perimeter of a rectangle, \(P = 2l + 2w\), 'l' and 'w' are variables representing the length and width, while '2' is a constant.
Understanding variables and their use is crucial for students as they allow us to describe and analyze real-world situations mathematically. This includes everything from calculating financial outcomes to understanding scientific data. In our plane traveling problem, the variable 'r' represents the rate or speed of the plane, which is crucial for determining the travel time.

Writing Algebraic Expressions

Writing algebraic expressions is a fundamental skill in algebra. It involves translating a verbal description of a problem into a mathematical format using variables, numbers, and operation symbols. The main goal is to create a representation that captures the essence of a problem in algebraic terms, which can then be manipulated according to algebraic rules.

Process and Tips

• Identify the Variables: Determine what the variables represent in the problem. For instance, 'r' could stand for rate in a speed problem.
• Understand the Operations: Know what operations the words in the verbal description are implying, such as 'product of' meaning multiplication.
• Be Consistent: Use the same variable to represent the same quantity throughout an expression or set of problems.
• Keep it Simple: Start with the simplest expression and build up as necessary to avoid confusion.

In the context of the plane travel problem, we were asked to write an expression for the travel time based on the given rate of speed (r) and a fixed distance (200 miles). The expression \( T = \frac{200}{r} \) succinctly encapsulates the relationship between time, distance, and rate. As students practice writing these expressions, they develop the ability to translate complex problems into solvable algebraic equations.