One hour after a hiker left a camp, a cyclist set out to catch up. If the hiker traveled 3 miles per hour and the cyclist traveled 6 miles per hour faster, how long will it take the cyclist to catch up with the hiker? PLEASE HELP!!!!
The Correct Answer and Explanation is:
ChatGPT said:
✅ Correct Answer: 0.5 hours (or 30 minutes)
🔍 Step-by-Step Explanation:
We’re given:
- The hiker walks at 3 miles per hour.
- The cyclist sets off 1 hour later but travels 6 mph faster, so:
- Cyclist’s speed = 3 + 6 = 9 miles per hour
We need to find how long it takes the cyclist to catch up with the hiker after the cyclist starts.
🧠 Let’s Define a Variable:
Let t = the number of hours the cyclist travels before catching up.
Then during that same time:
- The cyclist travels a distance of:
distance = speed × time = 9 × t = 9tmiles - The hiker, however, has already walked for 1 hour more, so the hiker has been walking for t + 1 hours.
Hiker’s distance:distance = 3 × (t + 1) = 3(t + 1)miles
Since they meet when the distances are the same:Cyclist’s distance=Hiker’s distance\text{Cyclist’s distance} = \text{Hiker’s distance}Cyclist’s distance=Hiker’s distance9t=3(t+1)9t = 3(t + 1)9t=3(t+1)
🧮 Solve the Equation:
Distribute on the right side:9t=3t+39t = 3t + 39t=3t+3
Subtract 3t from both sides:6t=36t = 36t=3
Divide both sides by 6:t=36=0.5t = \frac{3}{6} = 0.5t=63=0.5
📏 Final Answer:
It takes the cyclist 0.5 hours, or 30 minutes, to catch up.
📘 Conceptual Explanation (Why This Works):
This problem is about relative speed and distance = rate × time. The hiker starts first, so when the cyclist begins, the hiker already has a head start. Since the cyclist is faster, he starts closing the gap at the difference in their speeds, which is 6 mph.
In one hour, the hiker walks 3 miles. Since the cyclist closes the gap at 6 miles per hour, it will take:Head startRelative speed=36=0.5 hours\frac{\text{Head start}}{\text{Relative speed}} = \frac{3}{6} = 0.5 \text{ hours}Relative speedHead start=63=0.5 hours
This checks out perfectly!
