Question

Use the following information. You yell across a canyon. Your sound travels at a rate of 343 meters per second. After you shout, it takes 4 seconds to hear your echo. How far are you from the canyon wall?

Short answer

The distance from you to the canyon wall is 686 meters.

Step-by-step solution

Understand the problem

You are trying to find the distance to the canyon wall from where you are. The sound, when yelled, first travels to the wall and then bounces back to reach your ears. The speed of sound is given as 343 meters per second and the total time taken for the sound to travel back to you after bouncing off the canyon wall is 4 seconds.

Divide the total time by 2

Since the total time of 4 seconds includes the time taken for the sound to travel both to the canyon wall and back, the time taken for the sound to travel to the canyon wall is actually half of this. Thus, the time that the sound took to reach the canyon wall is \(4 \, seconds \, / \, 2 = 2 \, seconds\).

Calculate the distance using the speed of sound

Now that we have the time it takes for the sound to travel to the canyon wall, we can calculate the distance using the speed-distance-time relationship: distance = speed * time. Therefore, the distance to the canyon wall is \(343 \, meters/second \, * \, 2 \, seconds = 686 \, meters\).

Key concepts

Speed-Distance-Time Relationship

Understanding the speed-distance-time relationship is crucial when solving problems related to motion. In its simplest form, the relationship is expressed as:
\[ \text{distance} = \text{speed} \times \text{time} \].
This equation implies that if you know any two of the quantities (speed, distance, time), you can calculate the third. It's a foundational concept used in various fields such as physics, engineering, and even in everyday life situations like planning a trip.

• Speed is how fast an object is moving and is typically measured in units like meters per second (m/s) or miles per hour (mph).
• Distance is how far an object travels and is measured in units such as meters (m) or miles (mi).
• Time is the duration over which the object is moving and is measured in seconds (s), minutes (min), or hours (h).

In the given exercise, to find the distance to the canyon wall, we multiply the speed of sound (343 m/s) by the time taken for the sound to travel to the wall and back (2 seconds, after dividing the total echo time by 2). This yields the distance of 686 meters.

Solving Real-World Problems

Practical problems can often be untangled by applying mathematical concepts. Such problems require understanding of the real-world context and deciding which mathematical methods to use. For instance, in our canyon echo scenario, you are facing a real-world problem that involves distance calculation using the speed of sound.

• Analyze the situation: Understand what is being asked, identify the given data, and consider which principles or formulas may apply.
• Formulate the problem in mathematical terms: In this case, recognizing the need for the speed-distance-time formula.
• Perform calculations: Apply the formula, and carefully carry out the arithmetic to find your answer.

Real-world problems like this one demonstrate the power and usefulness of mathematics in our everyday lives. Once you identify the correct approach, even complex situations can be simplified and resolved effectively.

Algebraic Thinking

Algebraic thinking involves the use of mathematical concepts and techniques to solve problems. It allows us to generalize patterns, analyze relationships, and think logically about numbers, shapes, and functions. In the context of the exercise, algebraic thinking is used to solve for the unknown distance by creating and manipulating an equation.

• Identify variables: Here, speed, distance, and time are the variables, with distance being the unknown we need to find.
• Set up an equation: Utilize the speed-distance-time relationship (\[ \text{distance} = \text{speed} \times \text{time} \]) to develop an equation based on the known variables (speed and time).
• Manipulate the equation if necessary: Since the sound travels the distance twice, we modify the time variable accordingly before applying it to the equation.

Through algebraic thinking, solving the problem becomes a systematic process, helping us to arrive at a logical solution.