To find the first time after 1 o'clock when the hands of a clock will form a 90° angle, we need to consider the positions of the hour and minute hands.

Step-by-Step Solution:

1. Hour Hand Movement: The hour hand moves 30° per hour (since there are 360° total and 12 hours). After 1 o'clock, at any time $t$ minutes after 1 o'clock, the hour hand will have moved: $\text{Hour hand angle} = 30^\circ + \frac{1}{2}t$ where $t$ is in minutes. This is because the hour hand moves 0.5° per minute.

2. Minute Hand Movement: The minute hand moves 6° per minute (since it moves 360° in 60 minutes). So at $t$ minutes after 1 o'clock, the minute hand will be at: $\text{Minute hand angle} = 6t$

3. Condition for 90° Angle: We want the absolute difference between the angles of the hour and minute hands to be 90°. Therefore, we set up the equation: $|(30 + 0.5t) - 6t| = 90$

   This gives us two possible cases:

   Case 1: $(30 + 0.5t) - 6t = 90$ Simplifying: $30 - 5.5t = 90$ $-5.5t = 60$ $t = \frac{60}{5.5} \approx 10.91 \text{ minutes}$

   Case 2: $6t - (30 + 0.5t) = 90$ Simplifying: $5.5t - 30 = 90$ $5.5t = 120$ $t = \frac{120}{5.5} \approx 21.82 \text{ minutes}$

Answer:

The first time after 1 o'clock that the hands of the clock form a 90° angle is approximately 10.91 minutes, which is around 1:11 AM.

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Related Questions:

1. When will the hands of the clock form a straight line after 1 o'clock?
2. How many times between 1 o'clock and 2 o'clock do the hands of a clock form a 90° angle?
3. What is the next time after 1:11 that the hands of a clock form a 90° angle?
4. How long after any hour will the hands of a clock overlap?
5. How often do the hands of a clock form a right angle during a 12-hour period?

Tip:

The hands of a clock align at a right angle (90°) exactly twice within each hour.