Question

A jet is flying nonstop from Baltimore, Maryland, to Jacksonville, Florida, at a speed \(r\) of 500 miles per hour. The distance \(d\) between the two cities is about 680 miles. Which equation models the number of hours \(t\) the flight will take? (A)\(t=\frac{680}{500}\) (B) \(t=680(500)\) (C) \(680=\frac{t}{500}\) (D) \(t=\frac{500}{680}\)

Short answer

The correct equation that models the number of hours the flight will take is (A) \(t = \frac{680}{500}\).

Step-by-step solution

Understand the Variables

In this problem, three variables are given. The speed \(r\) is 500 miles per hour, the distance \(d\) between the two cities is 680 miles, and the number of hours \(t\) the flight will take is what we are trying to find.

Apply the speed-distance-time formula

The speed-distance-time relationship states that the time taken (t) is equal to the distance (d) divided by the speed (r). You can use this relationship to set up an equation using the given variables, which results in \(t = \frac{d}{r}\).

Substitute the known variables

Now we substitute the known variables into this equation. The speed \(r\) is 500 miles per hour and the distance \(d\) is 680 miles. Therefore, the equation becomes \(t = \frac{680}{500}\).

Determine the correct answer

Looking at the options given in the exercise, the correct equation that models the number of hours \(t\) the flight will take is (A) \(t = \frac{680}{500}\). The other options do not accurately reflect the speed-distance-time relationship.

Key concepts

Algebra

When approaching problems involving speed, distance, and time, algebra plays a central role in constructing and solving equations based on the relationship between these variables. In algebra, we use symbols to represent variables, or unknown values, and we manipulate these symbols to solve for the variables. In the context of the provided exercise, the symbols used are \r for rate or speed, \r for distance, and \r for time.

Understanding how to translate a real-world situation into an algebraic expression is crucial for solving problems. For instance, the expression \r\( t = \frac{d}{r} \) is derived from the understanding that time is equal to distance divided by speed. This equation is an algebraic representation of the speed-distance-time relationship, and solving it allows us to determine one variable when the other two are known. In the given exercise, algebra simplifies a potentially complex concept into a manageable equation that is easy to solve.

Equations

Equations are fundamental tools in mathematics and are used to express the equivalence between two expressions. They consist of two parts separated by an equal sign, indicating that the expressions on both sides have the same value. In problems involving speed, distance, and time, equations allow us to represent the relationships between these quantities in a precise mathematical form.

Once an equation is established, such as \r\( t = \frac{d}{r} \) in our jet flying example, we can solve for the unknown variable by substituting the known values into the equation. This process gives us a numerical answer that makes sense in the context of the problem. For the Baltimore to Jacksonville flight, by using \r\( d = 680 \) miles and \r\( r = 500 \) miles per hour in the equation, we can solve for the time \r\( t \) it takes to complete the journey. The correct manipulation of equations is an essential skill in mathematics, as it provides a methodical way to reach solutions.

Problem Solving

Problem solving is a systematic process that involves understanding the problem, planning a method of solution, carrying out the plan, and evaluating the result. When faced with a mathematics problem, it's important to first comprehend what is being asked. This means identifying the known and unknown variables and deciphering the relevant relationships between them.

In our flight example, problem-solving skills are evident in recognizing that the key is to find the flight time \r\( t \), given the distance and speed. By applying the speed-distance-time relationship, a plan is formulated, which consists of setting up the correct equation. Following through with the plan involves substituting the known values into the equation and simplifying to find the answer. Finally, checking that the solution is reasonable in the context of the problem ensures that the problem-solving process was successfully completed. Through effective problem-solving techniques, students can tackle even the most challenging mathematical problems with confidence.