Problem 27 - Algebra for College Students

Question

Johnny leaves Memphis to visit his cousin Anne, who lives in the town of Hornsby, Tennessee, 80 mi away. He travels at an average rate of 50 mph. One- half hour later, Anne leaves to visit Johnny, traveling at an average rate of \(60 \mathrm{mph}\). How long after Anne leaves will it be before they meet?

Step-by-Step Solution

Answer

0.5 hours after Anne leaves

Step 1: Determine Johnny's travel time when Anne leaves

Johnny leaves Memphis and travels for half an hour before Anne starts her journey. To calculate the distance Johnny has traveled during this period, use the formula: Distance = Speed × Time. Given Johnny's speed is 50 mph and the time is 0.5 hours, the distance Johnny covers is:Distance = 50 mph × 0.5 hours = 25 miles.

Step 2: Calculate the remaining distance between Johnny and Anne

The total distance between Memphis and Hornsby is 80 miles. Johnny has already traveled 25 miles by the time Anne starts her journey. Therefore, the remaining distance is:Remaining distance = Total distance - Distance Johnny traveled = 80 miles - 25 miles = 55 miles.

Step 3: Determine the relative speed of the two travelers

Both Johnny and Anne are traveling towards each other. Johnny's speed is 50 mph and Anne's speed is 60 mph. Their combined speed is the sum of their individual speeds:Relative speed = Johnny's speed + Anne's speed = 50 mph + 60 mph = 110 mph.

Step 4: Calculate the time taken for them to meet

To find out how long it will take for them to meet once Anne starts traveling, use the formula: Time = Distance / Speed.The distance to cover is 55 miles at a relative speed of 110 mph:Time = 55 miles / 110 mph = 0.5 hours.

Key Concepts

Distance-Rate-Time Relationship

The distance-rate-time relationship is fundamental in understanding motion problems, including those involving relative speed. The formula is simple and powerful: \( \text{Distance} = \text{Rate} × \text{Time} \). This equation tells us how far something will go if we know how fast it's moving and for how long. For instance, in the exercise above, Johnny travels at 50 mph. By using the formula with a time of 0.5 hours, we can calculate that he travels 25 miles: \( 50 \text{ mph} × 0.5 \text{ hours} = 25 \text{ miles} \). Keep these relationships in mind, as they are key to solving many motion problems. Whenever you know two of these variables, you can always find the third.

Meeting Point Calculations

Calculating the meeting point between two moving objects helps us understand how far and how long each needs to travel before they cross paths. The calculation typically involves a few steps:

• First, determine how far each person has traveled by the time both are in motion. Here, Johnny has already moved 25 miles.
• Next, find the remaining distance between them. The total distance is 80 miles, leaving us with 55 miles after subtracting Johnny's distance.
• Lastly, use the relative speed and remaining distance to calculate the meeting time.

This step-by-step approach ensures you account for all variables and changes in motion before the travelers meet. In our problem, once Anne starts driving, the remaining distance is 55 miles, and we use the relative speed to find how long it'll take them to meet.

Relative Speed in Opposite Directions

When two objects travel toward each other, their relative speed is the sum of their individual speeds. Understanding this concept helps us quickly find out how fast two objects are closing the gap between them. For Johnny and Anne:

• Johnny's speed: 50 mph
• Anne's speed: 60 mph

Their combined speed, or relative speed, is: \( 50 \text{ mph} + 60 \text{ mph} = 110 \text{ mph} \). We use this combined speed to calculate the time it takes for them to meet. In our problem, with a remaining distance of 55 miles and a relative speed of 110 mph, we can find the meeting time using \( \text{Time} = \text{Distance} / \text{Speed} \). So, \( 55 \text{ miles} / 110 \text{ mph} = 0.5 \text{ hours} \). This means Anne and Johnny will meet half an hour after Anne begins her journey.