Between 4 and 5 o'clock, when the hands will be inclined at 60° for the first time?

Options

A. `10(10)/(11)` min past 4
B. `11(10)/(11)` min past 4
C. `12 ` min past 4
D. `9(10)/(11)` min . Past 4

Answer

To solve the problem of finding when the hands of the clock will be inclined at 60° for the first time between 4 and 5 o'clock, we can follow these steps:

### Step 1: Understand the positions of the clock hands
At 4 o'clock, the hour hand is at 120° (since each hour represents 30°: 4 hours × 30° = 120°). The minute hand is at 0°.

### Step 2: Set up the equation for the angle between the hands
Let the time be 4 hours and X minutes. The position of the minute hand after X minutes is given by:
\[ \text{Angle of minute hand} = 6X \text{ degrees} \]
(since the minute hand moves at 6° per minute).

The position of the hour hand after X minutes is given by:
\[ \text{Angle of hour hand} = 120 + \frac{X}{2} \text{ degrees} \]
(since the hour hand moves at 0.5° per minute).

### Step 3: Calculate the angle between the two hands
The angle between the hour and minute hands can be calculated using the formula:
\[ \text{Angle} = | \text{Angle of hour hand} - \text{Angle of minute hand} | \]
Substituting the expressions we derived:
\[ \text{Angle} = | (120 + \frac{X}{2}) - (6X) | \]
This simplifies to:
\[ \text{Angle} = | 120 - \frac{11X}{2} | \]

### Step 4: Set the angle equal to 60°
We want to find when this angle is 60°:
\[ | 120 - \frac{11X}{2} | = 60 \]

### Step 5: Solve the equation
This absolute value equation gives us two cases to consider:

**Case 1:**
\[ 120 - \frac{11X}{2} = 60 \]
Solving for X:
\[ 120 - 60 = \frac{11X}{2} \]
\[ 60 = \frac{11X}{2} \]
\[ 120 = 11X \]
\[ X = \frac{120}{11} \approx 10.91 \text{ minutes} \]

**Case 2:**
\[ 120 - \frac{11X}{2} = -60 \]
Solving for X:
\[ 120 + 60 = \frac{11X}{2} \]
\[ 180 = \frac{11X}{2} \]
\[ 360 = 11X \]
\[ X = \frac{360}{11} \approx 32.73 \text{ minutes} \]

### Step 6: Determine the first occurrence
Since we are looking for the first occurrence between 4 and 5 o'clock, we take the smaller value:
\[ X \approx 10.91 \text{ minutes} \]

### Conclusion
Thus, the hands of the clock will be inclined at 60° for the first time at approximately 4:10.9.

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