Distance Traveled by Sam and Nina: A Real World Application of Algebraic Equations

In a scenario similar to real-life travel and competition, Sam and Nina are traveling towards each other from two different towns. Sam leaves town A and heads towards town B at an average speed of 72 km/h, while Nina leaves town B and heads towards town A at an average speed of 88 km/h. The distance between the two towns is 37 kilometers. This article explores how to calculate the distance traveled by Sam when the two of them meet.

Calculating the Meeting Point

To solve this problem, we need to understand the concept of relative speed and use algebraic equations to find the meeting point.

Relative Speed

The relative speed of two objects moving towards each other is the sum of their individual speeds. Here, the relative speed is 72 km/h (Sam's speed) 88 km/h (Nina's speed) 160 km/h.

Time Taken to Meet

The time taken for Sam and Nina to meet can be calculated using the formula:

[ t frac{text{Distance}}{text{Relative Speed}} ]

Substituting the values:

[ t frac{37 text{ km}}{160 text{ km/h}} frac{37}{160} text{ hours} ]

Distance Traveled by Sam

The distance traveled by Sam can be calculated using the formula:

[ text{Distance traveled by Sam} text{Sam's speed} times t ]

Substituting the values:

[ text{Distance traveled by Sam} 72 text{ km/h} times frac{37}{160} text{ hours} ]

Performing the calculation:

[ text{Distance traveled by Sam} frac{72 times 37}{160} text{ km} frac{336}{160} text{ km} frac{21}{10} text{ km} 2.1 times 1.7 text{ km} 3.8 times 5.2 text{ km} 16.65 text{ km} ]

Thus, the distance traveled by Sam when they meet is 16.65 kilometers.

General Solution Using Equations

We can also find the meeting point using a more generalized approach with equations. Let Sam and Nina meet at a distance of x from town A after a time t.

Equations for Distance Traveled

For Sam, the distance traveled from A is x, and the equation representing this is:

[ text{Distance} text{Speed} times text{Time} implies x 72 text{ km/h} times t implies t frac{x}{72} ]

For Nina, the distance traveled from B is 37 - x, and the equation representing this is:

[ text{Distance} text{Speed} times text{Time} implies 37 - x 88 text{ km/h} times t implies t frac{37 - x}{88} ]

When they meet, the time taken by both is the same:

[ frac{x}{72} frac{37 - x}{88} ]

Solving for x:

[ 88x 72(37 - x) implies 88x 2664 - 72x implies 16 2664 implies x frac{2664}{160} implies x 16.65 text{ km} ]

Therefore, the distance traveled by Sam when they meet is 16.65 kilometers.

Conclusion

This problem showcases the practical application of algebraic equations in real-life situations. By understanding the concept of relative speed and using equations, we can efficiently solve problems related to the meeting point of two travelers moving towards each other from two different starting points.

Understanding such applications can help in various real-world scenarios, such as predicting when two vehicles might meet or planning travel routes.