Final answer:
The traffic signals will change simultaneously again 15 minutes after 9 AM, which is at 9:15 AM. This is found by calculating the least common multiple of the individual cycle times of the signals. The correct answer is (2) 9.15 AM.
Step-by-step explanation:
To determine when the traffic signals will change simultaneously again, we need to find the least common multiple (LCM) of their individual cycle times. The four signals change every 30 seconds, 60 seconds (1 minute), 45 seconds, and 75 seconds, respectively.
Calculating the LCM of these numbers will give us the interval at which all signals will change together again. Here is the step-by-step explanation of how to find the LCM:
- List the prime factors of each number: 30 (2 × 3 × 5), 60 (2^2 × 3 × 5), 45 (3^2 × 5), and 75 (3 × 5^2).
- Identify the highest powers of prime factors from all the numbers: 2^2, 3^2, 5^2.
- Multiply these together to find the LCM: 2^2 × 3^2 × 5^2 = 4 × 9 × 25 = 900 seconds.
- Convert 900 seconds into minutes: 900 seconds ÷ 60 seconds/minute = 15 minutes.
The signals will change simultaneously again 15 minutes after 9 AM, which is 9:15 AM.