What is the first time after 1:00 that the hands make a 90 degree angle

To solve the problem of determining the first time after 1:00 when the hands of a clock form a 90-degree angle, we can use the concept of the relative speeds of the minute and hour hands.

### Step-by-step solution:
1. **Understanding how the clock hands move**:
   - The hour hand moves 0.5 degrees per minute (because it completes 360 degrees in 12 hours, or 30 degrees per hour).
   - The minute hand moves 6 degrees per minute (since it completes 360 degrees in 60 minutes).

2. **Angle between the hands**: 
   - After "t" minutes past 1:00, the hour hand will have moved \( 0.5t \) degrees from the 1:00 position (which starts at 30 degrees, since each hour represents 30 degrees).
   - The minute hand will have moved \( 6t \) degrees from the 12:00 position.

3. **Condition for a 90-degree angle**:
   - The angle between the two hands should be 90 degrees. The absolute difference in their positions must equal 90 degrees. This gives us the equation:
     \[
     |(30 + 0.5t) - 6t| = 90
     \]
     Simplifying this:
     \[
     |30 + 0.5t - 6t| = 90
     \]
     \[
     |30 - 5.5t| = 90
     \]
   - This gives two possible cases to solve:
     1. \( 30 - 5.5t = 90 \)
     2. \( 30 - 5.5t = -90 \)

### Solving both cases:

**Case 1**:
\[
30 - 5.5t = 90
\]
\[
-5.5t = 60
\]
\[
t = -\frac{60}{5.5} = -\frac{600}{55} = -\frac{120}{11} \approx -10.91 \text{ minutes}
\]
This gives a negative time, which is not physically meaningful in this context.

**Case 2**:
\[
30 - 5.5t = -90
\]
\[
-5.5t = -120
\]
\[
t = \frac{120}{5.5} = \frac{1200}{55} = \frac{240}{11} \approx 21.82 \text{ minutes}
\]

So, the first time after 1:00 that the hands form a 90-degree angle is approximately **1:21:49** (since 0.82 minutes is about 49 seconds).

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Would you like more details or have any questions? Here are 5 related problems you might find interesting:

1. What is the second time after 1:00 that the hands make a 90-degree angle?
2. At what time do the hands overlap after 1:00?
3. How often do the hands of a clock form a 180-degree angle?
4. Calculate the angle between the hands at 2:45.
5. What is the time between two consecutive 90-degree angles between the hands?

**Tip**: The hands of a clock form a 90-degree angle twice every hour.

What is the first time after 1:00 that the hands make a 90 degree angle?

Discover how to calculate the first time after 1:00 when the hands of a clock form a 90-degree angle. This problem explores clock geometry, the relative motion of hands, and key formulas for determining angles. Step-by-step solution for Grades 8-10.

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