can you help me?
fred leaves his house at 2pm and travels due north at 30km/hour. ron has been traveling due west at 20km/hour and arrives at fred's house at 3pm. where were they closest together?
Of course, I'd be happy to help you! To find out where Fred and Ron were closest together, we need to consider their positions at different times.
Let's start by finding the positions of Fred and Ron at 2 pm when Fred leaves his house:
Fred's position after t hours = t * speed
Fred's position at 2 pm = 2 * 30 = 60 km north (from his house)
Now let's find Ron's position at 2 pm. Since Ron arrives at Fred's house at 3 pm, he travels for 1 hour less than Fred:
Ron's position after t hours = t * speed
Ron's position at 3 pm = (3 - 1) * 20 = 2 * 20 = 40 km west (from Fred's house)
Now, let's plot their positions on a coordinate plane. Let's assume Fred's house is the origin (0, 0).
Fred's position at 2 pm: (0, 60)
Ron's position at 2 pm: (-40, 0)
To find where they were closest together, we need to find the shortest distance between these two points. We can use the Pythagorean theorem:
Distance^2 = (change in x)^2 + (change in y)^2
Change in x = -40 - 0 = -40
Change in y = 0 - 60 = -60
Thus, the distance^2 = (-40)^2 + (-60)^2 = 1600 + 3600 = 5200
To find the distance, we take the square root of 5200:
Distance = sqrt(5200) ≈ 72.11 km
Therefore, Fred and Ron were closest together approximately 72.11 km away from Fred's house.