Speed Time And Distance Mastering Problems For ENEM Airplane Travel Scenario

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In the vast realm of ENEM (Exame Nacional do Ensino Médio) preparation, mastering quantitative reasoning is paramount. Among the myriad topics tested, speed, time, and distance problems emerge as frequent contenders. These problems often involve scenarios that demand a keen understanding of the relationships between these variables and the ability to apply them in diverse contexts. This article dives deep into a classic speed, time, and distance problem, dissecting its core elements and providing a step-by-step solution. We'll also explore related concepts and strategies to equip you with the tools to conquer similar challenges on the ENEM.

Problem Statement

Consider this scenario: An airplane travels 960 kilometers against the wind at an average speed of (k-8) km per hour and then returns the same distance with the wind at an average speed of (k+8) km per hour. The trip against the wind took fifteen minutes longer than the trip with the wind. Calculate the value of k.

This problem presents a classic speed, time, and distance puzzle with an added twist – the wind's influence on the plane's speed. The key to unraveling this problem lies in understanding how the wind either adds to or subtracts from the plane's speed, depending on whether it's traveling with or against the wind.

Dissecting the Problem

Let's break down the problem into its fundamental components:

  • Distance: The airplane covers a fixed distance of 960 kilometers in both directions.
  • Speeds: The airplane's speed is affected by the wind. When flying against the wind, its effective speed is (k-8) km/h, and when flying with the wind, its effective speed is (k+8) km/h.
  • Time: The time taken for each leg of the journey differs. The trip against the wind takes 15 minutes longer than the trip with the wind. This is a crucial piece of information that will help us establish an equation.

Fundamental Concepts

Before we jump into the solution, let's revisit some fundamental concepts related to speed, time, and distance:

  • Speed: Speed is the rate at which an object moves, calculated as distance traveled per unit of time. It's typically measured in kilometers per hour (km/h) or meters per second (m/s).
  • Time: Time is the duration of an event or activity. It's commonly measured in hours, minutes, or seconds.
  • Distance: Distance is the length of the path traveled by an object. It's typically measured in kilometers or meters.

The Relationship

The relationship between speed, time, and distance is expressed by the following formula:

Distance = Speed × Time

This formula can be rearranged to solve for speed or time:

Speed = Distance / Time
Time = Distance / Speed

These formulas are the bedrock of solving speed, time, and distance problems. Understanding and applying them correctly is essential for success.

Solving the Problem: A Step-by-Step Approach

Now that we've dissected the problem and revisited the fundamental concepts, let's embark on solving it. Here's a step-by-step approach:

Step 1: Define Variables

Let's define the following variables to represent the unknowns in the problem:

  • t1 = Time taken to travel against the wind (in hours)
  • t2 = Time taken to travel with the wind (in hours)

Step 2: Formulate Equations

Using the formula Time = Distance / Speed, we can formulate equations for the time taken in each direction:

  • Time against the wind: t1 = 960 / (k - 8)
  • Time with the wind: t2 = 960 / (k + 8)

The problem states that the trip against the wind took 15 minutes (or 15/60 = 0.25 hours) longer than the trip with the wind. This gives us our third equation:

  • t1 = t2 + 0.25

Step 3: Substitute and Simplify

Now we have a system of three equations with three unknowns. We can substitute the expressions for t1 and t2 from the first two equations into the third equation:

960 / (k - 8) = 960 / (k + 8) + 0.25

To solve this equation, we need to eliminate the fractions. Multiply both sides of the equation by (k - 8)(k + 8):

960(k + 8) = 960(k - 8) + 0.25(k - 8)(k + 8)

Expanding the equation, we get:

960k + 7680 = 960k - 7680 + 0.25(k^2 - 64)

Step 4: Solve for k

Simplifying the equation further:

15360 = 0.25k^2 - 16

15376 = 0.25k^2

k^2 = 15376 / 0.25

k^2 = 61504

Taking the square root of both sides:

k = √61504

k = 248

Step 5: Verify the Solution

We've found that k = 248. To ensure our solution is correct, let's plug this value back into the original equations:

  • Speed against the wind: k - 8 = 248 - 8 = 240 km/h
  • Speed with the wind: k + 8 = 248 + 8 = 256 km/h
  • Time against the wind: t1 = 960 / 240 = 4 hours
  • Time with the wind: t2 = 960 / 256 = 3.75 hours

The difference in time is 4 - 3.75 = 0.25 hours, which is equal to 15 minutes. This confirms that our solution is correct.

Therefore, the value of k is 248.

Strategies for Tackling Speed, Time, and Distance Problems

Solving speed, time, and distance problems requires a strategic approach. Here are some tips to help you tackle these challenges effectively:

  • Read Carefully: Thoroughly read and understand the problem statement. Identify the given information and what you need to find.
  • Draw Diagrams: Visualizing the problem can often help. Draw diagrams to represent the scenario, including distances, speeds, and directions.
  • Define Variables: Assign variables to the unknown quantities. This will help you organize your thoughts and formulate equations.
  • Formulate Equations: Use the relationship between speed, time, and distance (Distance = Speed × Time) to set up equations. Look for clues in the problem statement that can help you establish relationships between the variables.
  • Solve the Equations: Solve the system of equations using algebraic techniques such as substitution or elimination.
  • Check Your Answer: Once you have a solution, plug it back into the original equations to verify its correctness. This will help you avoid careless errors.
  • Practice Regularly: The key to mastering speed, time, and distance problems is practice. Solve a variety of problems to develop your problem-solving skills and build confidence.

Common Variations and Extensions

Speed, time, and distance problems come in various forms. Here are some common variations and extensions you might encounter:

  • Relative Speed: Problems involving two objects moving towards or away from each other. The concept of relative speed is crucial here.
  • Average Speed: Problems asking for the average speed of an object over a journey with varying speeds. Remember that average speed is not simply the average of the speeds.
  • Circular Motion: Problems involving objects moving in a circular path. These problems often involve concepts like circumference and angular speed.
  • Upstream and Downstream: Problems similar to the one we solved, but involving boats traveling in rivers with currents. The current's speed affects the boat's speed in a similar way to the wind affecting the airplane's speed.

Conclusion

Mastering speed, time, and distance problems is essential for ENEM success. By understanding the fundamental concepts, practicing problem-solving strategies, and familiarizing yourself with common variations, you can confidently tackle these challenges. Remember, the key is to break down the problem, formulate equations, and solve them systematically. Keep practicing, and you'll be well on your way to conquering these types of problems on the ENEM.

This article has provided a comprehensive guide to solving a specific speed, time, and distance problem involving an airplane traveling with and against the wind. The step-by-step solution, along with the discussion of fundamental concepts and problem-solving strategies, will empower you to approach similar problems with confidence. Remember to practice regularly and apply these techniques to a variety of scenarios to solidify your understanding.

By diligently studying and practicing, you can master speed, time, and distance problems and significantly improve your performance on the ENEM. Good luck!