How many times between 2 p.m. and 5 pm does the angle between the hour hand and the minute hand become 90 degree
The correct answer and explanation is:
To calculate how many times the angle between the hour and minute hands becomes 90 degrees between 2 p.m. and 5 p.m., let’s break it down.
The angle between the hour and minute hands of a clock can be calculated using the formula: Angle=∣30H−5.5M∣\text{Angle} = \left| 30H – 5.5M \right|
Where:
- HH is the hour.
- MM is the minute.
We are looking for times when this angle is 90 degrees. Therefore, we want to solve the equation: ∣30H−5.5M∣=90\left| 30H – 5.5M \right| = 90
Step 1: At 2 p.m. (H = 2)
For 2 p.m., the equation becomes: ∣30(2)−5.5M∣=90\left| 30(2) – 5.5M \right| = 90 ∣60−5.5M∣=90\left| 60 – 5.5M \right| = 90
This gives two possible solutions:
- 60−5.5M=9060 – 5.5M = 90 leads to M=−5.45M = -5.45, which is not valid.
- 60−5.5M=−9060 – 5.5M = -90 leads to M=27.27M = 27.27.
So, at 2:27:16 p.m., the angle is 90 degrees.
Step 2: At 3 p.m. (H = 3)
For 3 p.m., the equation becomes: ∣30(3)−5.5M∣=90\left| 30(3) – 5.5M \right| = 90 ∣90−5.5M∣=90\left| 90 – 5.5M \right| = 90
This gives two possible solutions:
- 90−5.5M=9090 – 5.5M = 90 leads to M=0M = 0.
- 90−5.5M=−9090 – 5.5M = -90 leads to M=32.73M = 32.73.
So, at 3:00:00 p.m. and 3:32:43 p.m., the angle is 90 degrees.
Step 3: At 4 p.m. (H = 4)
For 4 p.m., the equation becomes: ∣30(4)−5.5M∣=90\left| 30(4) – 5.5M \right| = 90 ∣120−5.5M∣=90\left| 120 – 5.5M \right| = 90
This gives two possible solutions:
- 120−5.5M=90120 – 5.5M = 90 leads to M=5.45M = 5.45.
- 120−5.5M=−90120 – 5.5M = -90 leads to M=38.18M = 38.18.
So, at 4:05:27 p.m. and 4:38:11 p.m., the angle is 90 degrees.
Final Answer:
Between 2 p.m. and 5 p.m., the angle between the hour and minute hands becomes 90 degrees at 2:27:16 p.m., 3:00:00 p.m., 3:32:43 p.m., 4:05:27 p.m., and 4:38:11 p.m.
Thus, it happens five times during this period.