Problem 6 - Algebra 1: Concepts and Skills

Question

You are one mile from your home. You can walk at a speed of 4 miles per hour. The subway comes by every 15 minutes, and you heard one come by 3 minutes ago. The subway ride takes 8 minutes. Write a verbal model that relates the time it would take to walk home, your walking speed, and the distance to your home.

Step-by-Step Solution

Answer

The quickest way home is walking, which takes 15 minutes compared to the subway which takes 20 minutes.

Step 1: Calculate walking time

The walking speed is 4mph and the distance is 1 mile. The time taken to walk home is determined by dividing the distance by the speed. Thus: \( Time_{walk} = \frac{Distance}{Speed} = \frac{1 mile}{4 miles/hour} = 0.25 hours \). As there are 60 minutes in an hour, convert this to minutes: \( Time_{walk} = 0.25 hours * 60 = 15 minutes \)

Step 2: Calculate subway waiting and travel time

The problem states that the subway comes by every 15 minutes and one came by 3 minutes ago. Therefore, the wait time for the next subway is \( 15 minutes - 3 minutes = 12 minutes \). The subway ride itself takes 8 minutes. So, the total time for the subway option is: \( Time_{subway} = Wait_{subway} + Ride_{subway} = 12 minutes + 8 minutes = 20 minutes \)

Step 3: Compare times

Now compare each option's total time to find the quickest way home: Walking takes 15 minutes, and the subway takes 20 minutes. The quickest way is to walk.

Key Concepts

Distance-speed relationship

The relationship between distance, speed, and time is a fundamental concept in physics and mathematics. It helps us understand how quickly we can cover a particular distance at a certain speed. The core formula is:\[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]This equation tells us that the time required to travel a distance is inversely proportional to the speed. If you have a longer distance or a slower speed, it will take more time to cover that distance. Conversely, with a shorter distance or higher speed, you will cover it more quickly. In our exercise:

• The distance to home is 1 mile.
• The walking speed is 4 miles per hour.
• This gives a walking time of \( \frac{1}{4} \) hours or 15 minutes.

Understanding this relationship helps us make decisions based on how much time we have and the distance we need to travel.

Walking speed

Walking speed plays a crucial role in determining how long it takes to reach a destination by foot. It is measured in miles per hour (mph) or kilometers per hour (km/h). In our case, the walking speed is provided as 4 mph. When considering walking as a mode of transportation, it’s important to take into account how consistent the pace is and whether there are any obstacles along the way. For a straightforward distance of 1 mile:

• With a walking speed of 4 mph, it takes 0.25 hours to cover the distance.
• This translates to 15 minutes when converted from hours to minutes.

Knowing your walking speed allows you to plan trips accurately, ensuring that you arrive at your destination on time.

Subway schedule

Understanding the subway schedule is key to determining travel times when using public transportation. Subways often run on a fixed schedule, meaning they arrive at regular intervals. In our scenario, the subway arrives every 15 minutes. To calculate the wait time for the subway, you need to know when the last one left. Since it departed 3 minutes ago, the next one will arrive in 12 minutes:

• Subway wait time: 15 minutes - 3 minutes = 12 minutes.
• Subway ride itself takes an additional 8 minutes.
• Total subway travel time = 12 minutes waiting + 8 minutes riding = 20 minutes.

By understanding the schedule, you can decide whether it is more time-efficient to wait for the subway or choose another form of transportation.

Travel time comparison

Comparing travel times using different modes of transport is essential to choose the quickest option. In the given exercise, we compare walking versus taking the subway:

• Walking: Takes 15 minutes to walk 1 mile.
• Subway: Requires 12 minutes of waiting and 8 minutes of riding, totaling 20 minutes.

When faced with options, the best way to choose is to calculate the total time each option takes. Here, walking is faster than waiting for the subway and then taking the ride. This kind of comparison is useful in everyday situations where you need to weigh different factors like time, cost, and convenience before making a travel decision.