Final answer:
Julie and Sarah will catch up to each other after 8 meters, when they have both run for 1 hour at their respective speeds and taking into account their starting head starts.
Step-by-step explanation:
Julie and Sarah are in a race where Julie runs at 5 meters per hour and has a 3 meter head start, while Sarah runs at 3 meters per hour with a 5 meter head start. To find out how many meters they have to run before they catch up to each other, we can set up an equation based on their speeds and head starts.
Let x represent the time in hours it takes for them to meet. The distance Julie runs is given by her speed (5 meters/hour) times time (x) plus her head start (3 meters). Thus, Julie's distance can be represented as 5x + 3. Similarly, the distance Sarah runs is her speed (3 meters/hour) times time (x) plus her head start (5 meters), which is 3x + 5.
Since they meet at the same point, these distances must be equal: 5x + 3 = 3x + 5. Now let's solve for x:
- 5x + 3 = 3x + 5
- Subtract 3x from both sides: 2x + 3 = 5
- Subtract 3 from both sides: 2x = 2
- Divide by 2: x = 1
This means it takes 1 hour for Julie and Sarah to be at the same point. To calculate the distance, plug x = 1 into either equation. For Julie: 5(1) + 3 = 8 meters.
Therefore, Julie and Sarah will catch up to each other after they have both run 8 meters.