4. The hour and minute hands of a wall clock determine a right angle at exactly two moments between 11 and 12 o'clock. Determine, with second accuracy, how much time will pass between these two moments.
First solution.

Let $x$ be the number of minutes after 11 o'clock when the hour and minute hands will first form a right angle, i.e., an angle of $90^{\circ}$.

The minute hand moves $\frac{1}{60}$ of a full circle, or $\frac{360^{\circ}}{60}=6^{\circ}$, in one minute.

After $x$ minutes, the minute hand will have moved $(6 x)^{\circ}$.

The hour hand moves $\frac{360^{\circ}}{12}=30^{\circ}$ in one hour, so it moves $\frac{30^{\circ}}{60}=0.5^{\circ}$ in one minute. \quad 1 POINT

After $x$ minutes, the hour hand will have moved $(0.5 x)^{\circ}$.

The hour hand starts at the "11" position, and the minute hand starts at the "12" position, so the hands first form a right angle when

$30-0.5 x+6 x=90$,

2 POINTS

which simplifies to $5.5 x=60$, or

$x=\frac{600}{55}=\frac{120}{11}=10 \frac{10}{11}$ minutes.

1 POINT

The hands will first form a right angle at 11:10 and $\frac{10}{11}$ minutes.

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-23.jpg?height=448&width=384&top_left_y=907&top_left_x=1450)

Similarly, we determine when the hands will form a right angle for the second time. We have

$30-0.5 x+6 x=270$,

2 POINTS

which simplifies to $5.5 x=240$, or

$x=\frac{2400}{55}=\frac{480}{11}=43 \frac{7}{11}$ minutes.

1 POINT

The hands will form a right angle for the second time at 11:43 and $\frac{7}{11}$ minutes.

Thus, the time between these two moments is $43 \frac{7}{11}-10 \frac{10}{11}=32 \frac{8}{11}$ minutes.

1 POINT

One minute has 60 seconds, so $\frac{8}{11} \text{ min}=\frac{8}{11} \cdot 60 \text{ s} \approx 44 \text{ s}$. In the period between 11 and 12 o'clock, approximately 32 minutes and 44 seconds will pass between the two desired moments.