document.write( "Question 1162693: On a 12-hour clock, at what precise times (in fractions of a minute or second) would the hands be at right angles (90 degrees) between 10 and 11 o'clock? \n" ); document.write( "
Algebra.Com's Answer #786533 by Theo(13342)\"\" \"About 
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the hour hand is 300 degrees at 10:00 and 330 degrees at 11:00
\n" ); document.write( "there are 60 minutes between 10:00 and 11:00
\n" ); document.write( "therefore, the hour hand moves .5 degrees every minute.
\n" ); document.write( "10:00 is 300 degrees.
\n" ); document.write( "11:00 is 60 minutes later * .5 = 30 degrees more = 330 degrees.
\n" ); document.write( "this part checks out.
\n" ); document.write( "the minute hand moves 6 degrees every minute.
\n" ); document.write( "60 * 6 = 360 degrees.
\n" ); document.write( "this part checks out.
\n" ); document.write( "you want the difference between the hour hand position and the minute hand position to be either 90 degrees or 270 degrees.
\n" ); document.write( "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.
\n" ); document.write( "you start the hour hand at 10:00 which means the hour hand is at the 300 degree position.
\n" ); document.write( "you start the minute hand at 12:00 which means the minute hand is at the 360 degree position.
\n" ); document.write( "you want the minute hand position minus the hour hand position to be equal to 90 degrees or 270 degrees.
\n" ); document.write( "let x = the number of degrees the hour hand moves in a minute
\n" ); document.write( "then 12x = the number of degrees the minute hand moves in a minute.
\n" ); document.write( "your formula becomes:
\n" ); document.write( "d = 360 + 12x - (300 + x)
\n" ); document.write( "simplify to get:
\n" ); document.write( "d = 360 + 12x - 300 - x
\n" ); document.write( "combine like terms to get:
\n" ); document.write( "d = 60 + 11x
\n" ); document.write( "when d = 90, the formula becomes:
\n" ); document.write( "90 = 60 + 11x
\n" ); document.write( "subtract 60 from both sides to get:
\n" ); document.write( "30 = 11x
\n" ); document.write( "solve for x to get:
\n" ); document.write( "x = 30/11
\n" ); document.write( "when d = 270, the formula becomes:
\n" ); document.write( "270 = 60 + 11x
\n" ); document.write( "subtract 60 from both sides to get:
\n" ); document.write( "210 = 11x
\n" ); document.write( "solve for x to get:
\n" ); document.write( "x = 210/11
\n" ); document.write( "when x = 30/11, you get the following:
\n" ); document.write( "for the hour hand, the number of degrees becomes 300 + 30/11 = 302.7272727
\n" ); document.write( "for the minute hand, the number of degrees becomes 360 + 12 * 30/11 = 392.7272727
\n" ); document.write( "the difference is 90 degrees.
\n" ); document.write( "when x = 210/11, you get the following:
\n" ); document.write( "for the hour hand, the number of degrees becomes 300 + 210/11 = 319.0909091
\n" ); document.write( "for the minute hand, the number of degrees becomes 360 + 12 * 210/11 = 589.0909091
\n" ); document.write( "the difference is 270 degrees.
\n" ); document.write( "for 90 degrees difference, the hour hand is at 302.7272727 degrees.
\n" ); document.write( "in terms of hours, the hour is therefore 302.7272727 / 360 * 12 = 10.09090909
\n" ); document.write( "that's roughly 10 o'clock and 5 minutes and 27 seconds.
\n" ); document.write( "that would be 10:05:27.
\n" ); document.write( "this is confirmed by the picture of the clock at 90 degrees which shows the time as 10:05:25.
\n" ); document.write( "
\n" ); document.write( "for 270 degrees difference, the hour hand is at 319.0909091 degrees
\n" ); document.write( "in terms of hours, the hour is therefore 319.0909091 / 360 * 12 = 10.63636364.
\n" ); document.write( "that's roughly 10 o'clock and 38 minutes and 11 seconds.
\n" ); document.write( "this is confirmed by the picture of the clock at 270 degrees which shows the time as 10:38:11
\n" ); document.write( "
\n" ); document.write( "at 90 degree difference, your solution is:
\n" ); document.write( "10:05.454545455 or 10:05:27.27272727.
\n" ); document.write( "at 270 degree difference, your solution is:
\n" ); document.write( "10:38.18181810 or 10:38:10.90909091
\n" ); document.write( "the first form of the answer is in fractions of minutes.
\n" ); document.write( "the second form of the answer is in fractions of seconds.
\n" ); document.write( "both forms are based on clock times.\r
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