Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Travel Time Two friends fly from Denver to Orlando (a distance of 1526 miles). It takes 1 hour and 15 minutes to fly the first 500 miles. At this rate, how long is the entire flight?

Step-by-Step Solution

Answer

The total flight time from Denver to Orlando is approximately 3 hours and 49 minutes.

Step 1: Determine the rate

First, we need to convert the given time of the first leg of the trip, which is 1 hour and 15 minutes to minutes only for simplicity. This equals to 75 minutes. The rate of speed of the friends can be calculated using the formula speed = distance/time. For this leg of the flight, it turns out to be \( \frac{500}{75} \) or around 6.67 miles per minute.

Step 2: Calculate the total time

Now with this rate, the total time for the entire flight (1526 miles) can be calculated by rearranging the formula to solve for time. This gives us \( \frac{1526}{6.67} \), which equals approximately 229 minutes.

Step 3: Convert minutes to hours and minutes

Finally, we convert the total time, which is currently in minutes, back into hours and minutes for a more understandable result. To do this, we divide 229 by the number of minutes in an hour (60). This gives us 3.816 recurring, which can be rounded to 3.82. To obtain the number of minutes, we calculate the decimal part, 0.82, as a fraction of an hour. So, 0.82 * 60 = approx. 49 minutes. Therefore, the total flight time is around 3 hours and 49 minutes.

Key Concepts

Distance Rate Time Formula

One of the most fundamental concepts in travel time calculations is the Distance Rate Time Formula. This equation helps to relate distance, rate (or speed), and time in a straightforward mathematical expression. The formula is represented as:

• Distance = Rate × Time
• Rate = Distance ÷ Time
• Time = Distance ÷ Rate

In the context of our exercise, we want to find the rate or speed at which the friends are traveling during the first leg of the trip. First, we convert the flight leg's details into a format suitable for the formula application. By knowing the distance (500 miles) and time (75 minutes), we use the formula: \[ \text{Rate} = \frac{\text{Distance}}{\text{Time}} = \frac{500}{75} \]This computation, yielding approximately 6.67 miles per minute, tells us how quickly the friends are traveling initially. With this rate known, we can predict travel times for the entire journey.

Unit Conversion

To correctly use and apply mathematical formulas, especially in travel scenarios, proper unit conversion is vital. Units must match across different values to perform calculations without errors. In the exercise, one crucial conversion involves time. The initially given time is 1 hour and 15 minutes. To manage it with the distance-rate-time equation, all time measurements must be in minutes. We convert hours into minutes realizing that:

• 1 hour = 60 minutes
• 1 hour and 15 minutes = 75 minutes

This ensures accurate calculation of rate and allows us to only deal with one type of unit consistently, which simplifies future steps. Converting back to more useful time formats (hours and minutes) serves to present final results in a more familiar context.

Flight Duration Calculation

After obtaining the rate of travel (6.67 miles per minute) from the first leg of the flight, we can proceed to calculate the entire flight duration. This involves using the Distance Rate Time formula again: \[ \text{Total Time} = \frac{\text{Total Distance}}{\text{Rate}} = \frac{1526 \, \text{miles}}{6.67 \, \text{miles/minute}} \]This calculation gives a result of approximately 229 minutes for the whole distance of 1526 miles. However, expressing this time solely in minutes isn't the most convenient since flights are typically tracked in hours and minutes. To convert:

• Divide the total minutes by 60 to find hours: 229 minutes ÷ 60 = 3.816 recurring hours.
• Determine hours and the leftover in minutes: 3 hours and approximately 49 minutes.

So, the full flight duration is about 3 hours and 49 minutes, providing a practical perspective for travelers planning their itineraries.