Problem 53 - Algebra A Combined Function

Question

A Japanese "bullet" train set a new world record for train speed at 552 kilometers per hour during a manned test run on the Yamanashi Maglev Test Line in April 1999. The Yamanashi Maglev Test Line is 42.8 kilometers long. How many minutes would a test run on the Yamanashi Line last at this record- setting speed? Round to the nearest hundredth of a minute.

Step-by-Step Solution

Answer

The test run would last approximately 4.65 minutes.

Step 1: Understand the Problem

The problem requires us to find out how many minutes it takes for the train to travel the Yamanashi Maglev Test Line, which is 42.8 kilometers long, at a speed of 552 kilometers per hour.

Step 2: Convert the Speed to Kilometers per Minute

Since the train's speed is given in kilometers per hour, we first need to convert it to kilometers per minute for easier calculation with time in minutes. The speed is 552 kilometers per hour.To convert this to kilometers per minute, divide by 60 (since there are 60 minutes in an hour):\[ \text{Speed in km/min} = \frac{552}{60} = 9.2 \text{ km/min} \]

Step 3: Calculate Time in Minutes

Now that we have the train's speed in kilometers per minute, we can calculate the time it takes to travel 42.8 kilometers.Time is calculated using the formula: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).Substitute the values:\[ \text{Time} = \frac{42.8 \text{ km}}{9.2 \text{ km/min}} = 4.6522 \text{ minutes} \]

Step 4: Round the Time to the Nearest Hundredth

The calculated time is 4.6522 minutes. We round it to the nearest hundredth:\[ \text{Rounded Time} = 4.65 \text{ minutes} \].

Key Concepts

Rate of Speed Conversion

When you want to determine how long it will take to travel a certain distance, but your speed is given in different units such as kilometers per hour, it's helpful first to convert this speed into a more usable unit like kilometers per minute. This is because you're often calculating time in minutes.
To convert kilometers per hour to kilometers per minute, you'll divide by 60, since there are 60 minutes in an hour. This makes the units compatible for calculations that involve minutes.
The formula is:

• Speed in km/min = (Speed in km/hour) ÷ 60
• For example, if a train moves at 552 km/h, converting this to km/min would be:
• Speed in km/min = 552 ÷ 60 = 9.2 km/min

This conversion simplifies time calculations over short distances, making it easier to work out precisely how long a trip will last.

Distance-Time Calculation

Once you have your speed in kilometers per minute, you can easily calculate how long it will take to cover a certain distance using a simple formula. The formula for time, based on distance and speed, is:

• Time = Distance ÷ Speed
• In our example with the Japan's "bullet" train on the Yamanashi Maglev Test Line, the distance is 42.8 kilometers.

With the train speed already converted into kilometers per minute, you substitute into the formula:

• Time = 42.8 km ÷ 9.2 km/min
• This results in around 4.6522 minutes.

This calculated time gives a detailed understanding of how long the travel will take at the specified speed.

Rounding Decimals

After computing the time taken for travel in minutes, you might end up with a number that isn't very neat or easy to interpret. Therefore, rounding decimals is an essential step in giving a more practical answer.
Rounding to the nearest hundredth involves looking at the third decimal place.

• If it's 5 or more, you round up the second decimal.
• If it's less than 5, you keep the second decimal as it is.

In the bullet train example, the time was 4.6522 minutes. Here's how you round it:

• The third decimal is a 2, which is less than 5.
• Thus, the second decimal stays the same.
• 20So, 4.6522 rounds to 4.65 minutes.

Rounding makes numbers easier to read and use, especially when communicating your final results.