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How many times in a day do the hour and minute hands of a correctly running clock form an angle of $90^{\circ}$?

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The angle between the hour and minute hands changes uniformly and continuously.

## Solution

Let midnight (the start of the day) be the initial moment of time \( t = 0 \). In one hour, the minute hand completes one full rotation, i.e., \( 360^\circ \), while the hour hand completes \( 30^\circ \). At time \( t \) (where \( t \) is measured in hours), the total rotation of the minute hand is \( M(t) = 360t^\circ \), and the total rotation of the hour hand is \( C(t) = 30t^\circ \). The angle between the hour and minute hands is \( 90^\circ \) if and only if the value \( M(t) - C(t) \) is equal to \( 90 + 180n^\circ \) for some integer \( n \). However, \( M(t) - C(t) = 330t^\circ \). Over the course of a day, \( t \) can vary from 0 to 24, so the value \( M(t) - C(t) \) continuously increases from 0 to \( 330 \times 24^\circ \). Therefore, the values of the form \( 90 + 180n^\circ \) are taken by the function \( M(t) - C(t) \) for \( n = 0, 1, \ldots, 43 \) (since \( 90 + 180 \times 43 < 330 \times 24 = 180 \times 44 < 90 + 180 \times 44 \)). Thus, the right angle between the hands will occur 44 times in a day.

## Answer

44.00