one elevator leaves first floor of a building at 9:00 a.m. It takes 5 seconds to travel between floors, stays 10 seconds at each floor and stops at each floor on its way to the top. Another elevator leaves the 12th (top) floor at 9:00 a.m. ; it takes 5 seconds to travel between floors, stays 15 seconds at each floor and stops at each floor on its way to the bottom. On what floor will both elevators be available at the same time?
time floor
0 1
0-5 1-2
5-15 2
15-20 2-3
20-30 3
30-35 3-4
35-45 4
45-50 4-5
50-60 5
60-65 5-6
65-75 6
75-80 6-7
80-90 7
90-95 7-8
0 12
0-5 11-12
5-20 11
20-25 10-11
25-40 10
40-45 9-10
45-60 9
60-65 8-9
65-80 8
80-85 7-8
85-100 7
for t 85-90 both are on floor 7
To determine on what floor both elevators will be available at the same time, we need to find the least common multiple (LCM) of their stopping intervals.
The first elevator takes 5 seconds to travel between floors and stays 10 seconds at each floor, which means it takes a total of 15 seconds between each floor (5 seconds for travel + 10 seconds of stay).
The second elevator takes 5 seconds to travel between floors and stays 15 seconds at each floor, resulting in a total of 20 seconds between each floor (5 seconds for travel + 15 seconds of stay).
Now, we need to find the LCM of 15 seconds and 20 seconds to determine how long it will take for both elevators to align.
To find the LCM, we can list the multiples of each interval and find the smallest common multiple:
Multiples of 15 seconds: 15, 30, 45, 60, 75, 90, 105, ...
Multiples of 20 seconds: 20, 40, 60, 80, 100, 120, 140, ...
From the list, we can see that the smallest common multiple, or LCM, is 60 seconds.
Therefore, both elevators will be available at the same time every 60 seconds.
Now, let's calculate the floors at which both elevators will meet:
For the first elevator, it stops at each floor. Thus, it will take (12 floors) * (15 seconds per floor) = 180 seconds (or 3 minutes) to reach the top floor (12th floor).
For the second elevator, it also stops at each floor. So, it will take (12 floors) * (20 seconds per floor) = 240 seconds (or 4 minutes) to reach the bottom floor (1st floor).
Since both elevators start at the same time (9:00 a.m.) and meet every 60 seconds, you need to divide each travel time by 60 to determine which iteration will have both elevators available simultaneously.
For the first elevator: 3 minutes / 60 seconds = 3 iterations.
For the second elevator: 4 minutes / 60 seconds = 4 iterations.
Since both elevators are aligned every 60 seconds, and the second elevator will take 4 iterations to reach the first floor while the first elevator only takes 3 iterations to reach the top floor, both elevators will be available simultaneously at the (12th - 3) = 9th floor.
Therefore, both elevators will be available at the same time on the 9th floor.