Solve each problem. See Example 2. In the Apple Hill Fun Run, Mary runs at 7 mph, Janet at 5 mph. If they start at the same time, how long will it be before they are 1.5 mi apart?

Verified step by step guidance

1. Identify the speeds of Mary and Janet: Mary runs at 7 mph and Janet runs at 5 mph.
2. Determine the relative speed at which the distance between them increases by subtracting the slower speed from the faster speed: \$7 - 5 = 2$ mph.
3. Set up the distance formula relating distance, rate, and time: \(\text{distance} = \text{rate} \times \text{time}\).
4. Use the relative speed as the rate and 1.5 miles as the distance to find the time: \$1.5 = 2 \times t$.
5. Solve for \(t\) by dividing both sides of the equation by 2: \(t = \frac{1.5}{2}\).

Key Concepts

Relative Speed

Relative speed is the rate at which the distance between two moving objects changes. When two people move in the same or opposite directions, their relative speed is found by adding or subtracting their individual speeds. This helps determine how quickly the distance between them increases or decreases.

Distance-Speed-Time Relationship

The fundamental formula relating distance, speed, and time is Distance = Speed × Time. This equation allows us to calculate any one of these variables if the other two are known, which is essential for solving problems involving motion.

Problem Setup and Interpretation

Understanding how to set up the problem involves identifying what is given and what needs to be found. Here, knowing that Mary and Janet start simultaneously and move at different speeds helps us model their positions over time and find when their separation reaches 1.5 miles.