Question 1162693
the hour hand is 300 degrees at 10:00 and 330 degrees at 11:00
there are 60 minutes between 10:00 and 11:00
therefore, the hour hand moves .5 degrees every minute.
10:00 is 300 degrees.
11:00 is 60 minutes later * .5 = 30 degrees more = 330 degrees.
this part checks out.
the minute hand moves 6 degrees every minute.
60 * 6 = 360 degrees.
this part checks out.
you want the difference between the hour hand position and the minute hand position to be either 90 degrees or 270 degrees.
since the minute hand moves 6 degrees per minute and the hour hand moves .5 degrees per minute, then the minute hand moves 12 times as far as the hour hand each minute.
you start the hour hand at 10:00 which means the hour hand is at the 300 degree position.
you start the minute hand at 12:00 which means the minute hand is at the 360 degree position.
you want the minute hand position minus the hour hand position to be equal to 90 degrees or 270 degrees.
let x = the number of degrees the hour hand moves in a minute
then 12x = the number of degrees the minute hand moves in a minute.
your formula becomes:
d = 360 + 12x - (300 + x)
simplify to get:
d = 360 + 12x - 300 - x
combine like terms to get:
d = 60 + 11x
when d = 90, the formula becomes:
90 = 60 + 11x
subtract 60 from both sides to get:
30 = 11x
solve for x to get:
x = 30/11
when d = 270, the formula becomes:
270 = 60 + 11x
subtract 60 from both sides to get:
210 = 11x
solve for x to get:
x = 210/11
when x = 30/11, you get the following:
for the hour hand, the number of degrees becomes 300 + 30/11 = 302.7272727
for the minute hand, the number of degrees becomes 360 + 12 * 30/11 = 392.7272727
the difference is 90 degrees.
when x = 210/11, you get the following:
for the hour hand, the number of degrees becomes 300 + 210/11 = 319.0909091
for the minute hand, the number of degrees becomes 360 + 12 * 210/11 = 589.0909091
the difference is 270 degrees.
for 90 degrees difference, the hour hand is at 302.7272727 degrees.
in terms of hours, the hour is therefore 302.7272727 / 360 * 12 = 10.09090909
that's roughly 10 o'clock and 5 minutes and 27 seconds.
that would be 10:05:27.
this is confirmed by the picture of the clock at 90 degrees which shows the time as 10:05:25.
<img src = "http://theo.x10hosting.com/2020/072301.jpg" >
for 270 degrees difference, the hour hand is at 319.0909091 degrees
in terms of hours, the hour is therefore 319.0909091 / 360 * 12 = 10.63636364.
that's roughly 10 o'clock and 38 minutes and 11 seconds.
this is confirmed by the picture of the clock at 270 degrees which shows the time as 10:38:11
<img src = "http://theo.x10hosting.com/2020/072302.jpg" >
at 90 degree difference, your solution is:
10:05.454545455 or 10:05:27.27272727.
at 270 degree difference, your solution is:
10:38.18181810 or 10:38:10.90909091
the first form of the answer is in fractions of minutes.
the second form of the answer is in fractions of seconds.
both forms are based on clock times.