clock problem:
how soon after 1 o clock will the hands of a clock form a right angle
the minute hand rotates once per hour, so it moves 360/60 = 6°min
The hour hand takes 12 hours to go around, so it moves 360/720 = 1/2 °/min
So you want t minutes, such that
0 + 6t = 30 + 1/2 t + 90
Now solve for t
Consider noon to correspond with 0° (standard "bearing")
and consider the 1:00 position.
The angle formed by the hour hand between noon and 1:00 is 30°
and the angle formed by the minute hand is 0°
after a time covered by x°, the hour hand is at 30+x/60 °
and the minute hand is at x °
But we want
x - (30+x/60) = 90
59x/60 = 120
x = 122.0339 °
so the minute hand is at the 122.0339° position
122.0339/360 = minute/60
minutes = 20.339 or 20 minutes, 20 seconds
which puts the time at 1:20:20
check my arithmetic
To find out how soon after 1 o'clock the hands of a clock will form a right angle, we need to understand the concept of the angle between the hour hand and the minute hand.
At 1 o'clock, the minute hand points at the 12 on the clock, and the hour hand points at the 1 on the clock. Let's assume that the hour hand moves at a constant speed while the minute hand moves continuously.
To form a right angle, the hour hand and the minute hand need to be exactly 90 degrees apart.
The minute hand moves 360 degrees in 60 minutes (1 complete rotation in 1 hour). Therefore, in 1 minute, it moves 6 degrees (360 degrees divided by 60 minutes).
The hour hand moves 30 degrees in 60 minutes (1 complete rotation in 12 hours). Therefore, in 1 minute, it moves 0.5 degrees (30 degrees divided by 60 minutes).
To form a right angle, the minute hand needs to be ahead of the hour hand by 90 degrees.
Now, let's calculate how long it takes for the minute hand to be exactly 90 degrees ahead of the hour hand:
90 degrees / (6 degrees per minute - 0.5 degrees per minute) = 90 / 5.5 ≈ 16.3636 minutes
Therefore, the hands of the clock will form a right angle approximately 16.36 minutes after 1 o'clock.
To determine how soon after 1 o'clock the hands of a clock will form a right angle, we need to understand the relationship between the hour hand and the minute hand.
The important thing to know is that while the minute hand moves continuously around the clock, the hour hand moves in increments from one hour to the next. In other words, the hour hand jumps from one hour to the next as the minute hand completes a full rotation.
To find the time when the hour and minute hands form a right angle, we can follow these steps:
Step 1: Calculate the angle covered by the hour hand in one minute.
- In 60 minutes, the hour hand moves 1/12th of the way between two neighboring hours.
- Since there are 360 degrees in a circle and 12 hours on a clock, the angle covered by the hour hand in one minute is 360 degrees / (12 hours x 60 minutes) = 0.5 degrees.
Step 2: Calculate the angle covered by the minute hand in one minute.
- In 60 minutes, the minute hand completes a full rotation of 360 degrees.
- Therefore, the angle covered by the minute hand in one minute is 360 degrees / 60 minutes = 6 degrees.
Step 3: Calculate the difference in angles between the hour and minute hands.
- Since the minute hand moves at a faster rate, the hour hand lags behind. Thus, the difference in angles between the hour and minute hands decreases by 0.5 degrees every minute (as calculated in step 1 and step 2).
- We need to find the point at which the difference is exactly 90 degrees.
Step 4: Calculate the time when the hands of a clock form a right angle.
- Start at 1 o'clock and calculate the difference in angles between the hour and minute hands for each minute until the difference reaches 90 degrees.
- The time when the hands of the clock form a right angle is the result.
Using these steps, you can work through the calculations to find the exact time when the hands of the clock form a right angle after 1 o'clock.