Question

at what time between 8:00 and 9:00 will the hands of clock be at right angle​

Answer 1

Answer:

When the hour hand is between 8 and 9 there are two positions of the minute hand when the hour hand and the minute hand form right right angles with each other. At 9 o’clock the two hands are at 90 degrees to each other. If that is unacceptable it will be between 8:25 and 8:30. What will be the exact time can be worked out as follows:

While the minute hand turns through 360 degrees in an hour, the hour hand covers 30 degrees during the same period. At 8:25 the minute hand would have covered 360*25/60 = 150 degrees. At that time the hour hand would have covered 30*25/60 = 12.5 degrees or 30*8+12.5 = 252.5 degrees from the 12 o’clock position. Thus the two hands are 252.5 - 150 = 102.5 degrees apart

At 8:30 the minute hand would have covered 360*6/12 = 180 degrees. At that time the hour hand would have covered 30*6/12 = 15 degrees or 30*8+15 = 255 degrees from the 12 o’clock position. Thus the two hands are 255 - 180 = 75 degrees apart.

Let the exact time when the two hands are 90 degrees to each other be x from the 12 o’clock position. At that time the minute hand would have covered 360*x/60 = 60 x degrees.

At the same time the hour hand would have covered 30*x/5 +8*30= 6 x +240 degrees.

6 x +240 - 60 x = 90

or 150 = 54 x or x = 150/54 = 2.777

So at 8:2.777 or 8:27:46.67 the two hands will form 90 degrees at the center of the dial.

Answer 2

Given,

The starting time$\[ = 8:00\,{\rm{am}}\]$

The end time$\[ = 9:00\,{\rm{am}}\]$

Solution,

Know that the minute hand covers $\[360^\circ \]$ in one hour and in the same time the hour hand covers $\[30^\circ \]$.

Assume that when the two hands are $\[90^\circ \]$ to each other the position from twelve o clock be y.

Therefore,

$\[\begin{array}{l} \Rightarrow 30 \times \frac{y}{5} - 8 \times 30 = 90\\ \Rightarrow 6y - 240 = 90\\ \Rightarrow y = 2.77\end{array}\]$

Hence, the exact time is $\[8:27\,{\rm{am}}.\]$