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. Define the variable for the time they run before being 1.5 miles apart. Let this time be \(t\) hours.
2. Express the distance each person runs in terms of \(t\). Mary runs at 7 mph, so her distance is \(7 \times t\) miles. Janet runs at 5 mph, so her distance is \(5 \times t\) miles.
3. Since they start together and run in the same direction, the distance between them after time \(t\) is the difference of their distances: \(7t - 5t\) miles.
4. Set up the equation for the distance apart: \(7t - 5t = 1.5\) miles.
5. Simplify the equation to \(2t = 1.5\) and solve for \(t\) by dividing both sides 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 concept helps determine how quickly the distance between Mary and Janet increases.

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. In this problem, it helps find the time it takes for Mary and Janet to be 1.5 miles apart.

Problem Setup and Interpretation

Properly interpreting the problem involves understanding that both runners start simultaneously and move at constant speeds. Setting up the equation correctly by expressing the distance between them as a function of time is essential to solving for the time when they are 1.5 miles apart.