At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?

1. Break Down the Problem

We need to determine the time between 4 and 5 o'clock when the minute and hour hands of a clock are pointing in opposite directions (180 degrees apart).

2. Relevant Concepts

To find the positions of the clock hands:

• The hour hand moves at a rate of $30$ degrees per hour (since $360$ degrees / $12$ hours = $30$ degrees per hour).
• The minute hand moves at $6$ degrees per minute (since $360$ degrees / $60$ minutes = $6$ degrees per minute).

3. Analysis and Detail

At 4:00, the hour hand is at: $\text{Position of hour hand} = 4 \times 30 = 120 \text{ degrees}$ Let $m$ be the number of minutes past 4 o'clock.

Position of the hour hand at $4:m$: $\text{Position of hour hand at } 4:m = 120 + \frac{m}{2} \text{ degrees}$ (since $30$ degrees per hour = $\frac{1}{2}$ degrees per minute)

Position of the minute hand at $4:m$: $\text{Position of minute hand at } 4:m = 6m \text{ degrees}$

To find when the hands are opposite, we set up the equation: $\left| \left(120 + \frac{m}{2}\right) - 6m \right| = 180$

This creates two cases to solve:

1. $120 + \frac{m}{2} - 6m = 180$
2. $6m - \left(120 + \frac{m}{2}\right) = 180$

Solving Case 1:

$120 + \frac{m}{2} - 6m = 180$ $\frac{m}{2} - 6m = 180 - 120$ $\frac{m}{2} - 6m = 60$ Multiplying through by 2 to eliminate the fraction: $m - 12m = 120 \Rightarrow -11m = 120 \Rightarrow m = -\frac{120}{11} \text{ (not valid)}$

Solving Case 2:

$6m - \left(120 + \frac{m}{2}\right) = 180$ $6m - 120 - \frac{m}{2} = 180$ Combining like terms: $6m - \frac{m}{2} = 300 \Rightarrow \frac{12m - m}{2} = 300 \Rightarrow \frac{11m}{2} = 300$ Multiplying through by $2$: $11m = 600 \Rightarrow m = \frac{600}{11} \approx 54.55 \text{ minutes}$

4. Verify and Summarize

The approximate solution is $54.55$ minutes past 4. This corresponds to about $4:55$ and $\frac{5}{11}$ seconds.

Final Answer

The hands of the clock will point in opposite directions at approximately 4:54 and 5/11 minutes (or about 4:54:32).