Ladies' Diary/Hour and Minute Hand in Straight Line

Puzzle

 * "Being at so large a distance from the dial-plate of a great clock, that I could not distinguish the figures;
 * but as the hour and minute hands were very bright and glaring,"
 * the correspondent noted that they were in a straight line and pointing upwards to the right.
 * It was evening.


 * What was the time?

Solution

 * $5 \frac 5 {11}$ minutes, or approximately $5$ minutes and $27$ seconds, past $7$.

Proof
Let it be assumed from the description of the general direction of the hands, and the fact that it is evening, that it is some short time after $7$ p.m.

Thus the hour hand is pointing a little way past the $7$.

Hence the minute hand is pointing a little way past the $1$.

Let $x$ be the number of minutes after $7$ that it is.

Let $\theta$ be the angle made by the hands of the clock from the vertical at that time.

The minute hand rotates at $1$ revolution every $60$ minutes.

That is, $\dfrac {360 \degrees} {60} = 6 \degrees$ per minute.

The hour hand rotates at $1$ revolution every $12$ hours.

That is, $\dfrac {360 \degrees} {12 \times 60} = \dfrac 1 2 \degrees$ per minute.

But at $7$, the hour hand is already at $\dfrac {360 \degrees} {12} = 30 \degrees$ from the vertical.

Thus:

So it is $5 \frac 5 {11}$ minutes past $7$.

$\dfrac 5 {11}$ minutes is $\dfrac {5 \times 60} {11}$ seconds, which evaluates to $27.\dot 2 \dot 7$ seconds.