mr trocki is the groundskeeper for a vacation resort. he has programmed the sprinkler system to run at night in three different zones. The sprinklers turn on at midnight and run until 7am. the fist zone runs every 90 minutes, the second zone turns on every 40 minutes, and the third zone turns on every 45 minutes. write a work problem within the context of the vacation resort that can be solved by determining the GCF
At the vacation resort, Mr. Trocki, the groundskeeper, is responsible for maintaining the garden areas. One day, he noticed that the sprinkler systems for three different zones were turning on simultaneously sporadically throughout the night. This irregular timing was causing issues as the sprinklers were not working efficiently, resulting in uneven watering of the plants. To fix the problem, Mr. Trocki realized that he needed to synchronize the sprinkler system cycles in all three zones.
To find a solution, Mr. Trocki decided to determine the Greatest Common Factor (GCF) of the time intervals for each zone. The GCF would help him identify the smallest time duration that all the zones' sprinkler cycles could be divided evenly into. By doing so, he would be able to synchronize the sprinkler system effectively, ensuring that all the zones received equal watering.
Now, let's calculate the GCF based on the time intervals:
Zone 1: Sprinklers turn on every 90 minutes.
Zone 2: Sprinklers turn on every 40 minutes.
Zone 3: Sprinklers turn on every 45 minutes.
To find the GCF, we can list out the common factors of each zone and identify the largest value that they share:
Zone 1: 90, 180, 270, 360, 450, 540, 630, ...
Zone 2: 40, 80, 120, 160, 200, 240, 280, ...
Zone 3: 45, 90, 135, 180, 225, 270, ...
From the lists above, we can see that the largest common factor (GCF) is 90 minutes. Therefore, to synchronize the sprinkler system cycles, Mr. Trocki needs to adjust Zone 1 to run every 90 minutes instead of 90, Zone 2 to run every 90 minutes instead of 40, and Zone 3 to run every 90 minutes instead of 45. By doing so, all zones will align and turn on simultaneously every 90 minutes, ensuring proper and equal watering of the resort's garden areas.