Problem 47 - Intermediate Algebra

Question

Airport Walkways. A man walks at a steady pace as he steps onto a moving walkway. It takes him 40 seconds to reach the end, 320 feet away. If he walks at the same rate against the flow of the walkway, it would take him 80 seconds to reach the end. Find his rate of walking and the rate of the moving walkway. (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The man's walking speed is 6 feet per second, and the walkway speed is 2 feet per second.

Step 1: Define Variables

Let's define two variables to make the problem more manageable. Let \( r_w \) be the man's walking speed in feet per second, and let \( r_m \) be the speed of the moving walkway in feet per second.

Step 2: Set Up Equations for Speeds

When the man walks with the moving walkway, his combined speed is \( r_w + r_m \). When he walks against the walkway, his effective speed is \( r_w - r_m \).

Step 3: Create Equations for Time and Distance

We know the distance is 320 feet. For the walk with the walkway, he takes 40 seconds, so the equation is \( (r_w + r_m) \times 40 = 320 \). For the walk against the walkway, he takes 80 seconds, giving the equation \( (r_w - r_m) \times 80 = 320 \).

Step 4: Simplify and Solve the System of Equations

Simplify the equations:\(* \) With the walkway: \( r_w + r_m = 8 \).\(* \) Against the walkway: \( r_w - r_m = 4 \).\(\) Add the equations: \( 2r_w = 12 \), so \( r_w = 6 \).\(\) Substitute \( r_w = 6 \) into the first equation: \( 6 + r_m = 8 \), so \( r_m = 2 \).

Step 5: Check the Solution

Substitute \( r_w = 6 \) and \( r_m = 2 \) back into both original equations to verify the correctness: \( (6 + 2) \times 40 = 320 \) is correct, and \( (6 - 2) \times 80 = 320 \) also holds true.

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. In the context of our walkway problem, we are dealing with two equations involving two variables, which are the man's walking speed and the walkway's speed.
When approaching a problem like this:

• Identify what you need to find. Here, it's the walking speed and the walkway speed.
• Define variables for each unknown component. For example, let \( r_w \) represent walking speed and \( r_m \) be the walkway speed.
• Establish equations based on given information. For the walkway problem, one equation considers the man's speed with the walkway, and the other against it.
• Solve the system using methods such as substitution or elimination. By working through these equations, you can determine the unknown values.

Systems of equations allow us to break down complex problems into smaller, more manageable parts. This technique is widely applicable, from physics to economics, making it a valuable tool for problem-solving.

Distance-Speed-Time Problems

Distance-speed-time problems often involve figuring out one or more of these quantities given the others. The core relationship here is \( \text{Distance} = \text{Speed} \times \text{Time} \). In our exercise:

• We consider two scenarios: moving with the walkway and against it.
• The walkway enhances the man's speed when moving with it, whereas it diminishes his speed when moving against it.
• For the first scenario, the equation becomes \( (r_w + r_m) \times 40 = 320 \).
• In the second, it's \( (r_w - r_m) \times 80 = 320 \).

The key here is understanding how different speeds come into play and using the equations to solve for unknowns based on given distances and times. Mastering these problems enhances one's ability to think critically about moving objects and their interactions.

Problem Solving Strategies

Tackling algebra word problems can initially seem daunting, but following a structured approach can make this easier. Here are some strategies:

• Read Carefully: Understand what is being asked. Identify knowns and unknowns.
• Define Variables: Assign symbols to unknown quantities to simplify thinking and expression.
• Translate Words into Equations: Turn the problem statement into mathematical expressions. This is crucial in bridging the real-world scenario to solvable math problems.
• Solve Step-by-Step: Work through the problem methodically, simplifying as you go. Deal with one part of the problem at a time.
• Check your Work: Once you have a solution, plug your values back into the original problem. Ensure all conditions of the problem are met.

By applying these strategies, students can develop a systematic way to solve complex problems, enhancing both confidence and mathematical acumen.