Depth · one concept from first principles · step 2 of 4
A clock shows a time between o'clock and o'clock. Exactly minutes from now, the hour hand and the minute hand will point in exactly opposite directions, forming a straight line. What time is it now?
At m minutes past 9, the hour hand starts at 270 degrees and turns 1/2 degree a minute; the minute hand turns 6 degrees a minute from 12.
Each hand is just a starting angle plus a rate times the minutes.
7.RP.A.1Draw A DiagramThe difference of the two angles is 180 either way round.
This is the case where the hour hand is still ahead of the minute hand.
7.EE.B.4Convert To AlgebraThe hands point opposite ways when the hour hand is degrees ahead of the minute hand, which happens minutes after 9.
Pointing in opposite directions means the two hands make one straight line through the center, a straight angle.
Each hand's angle is its starting angle plus its turning speed times the minutes.
The minute hand turns all the way round in minutes, so it turns degrees each minute.
All the way round the dial is one full turn, degrees.
At a steady speed, degrees in minutes is degrees per minute.
The hour hand moves one number, degrees, in minutes, so degree per minute, starting from the 9 at degrees.
Both hands turn the same way, so the gap shrinks by degrees a minute, and closing degrees takes minutes.
If the minute hand were 180 degrees ahead instead, m = 900/11, which is over 60, so it is past 10 o'clock and not allowed.
The time must stay within this hour, which rules this case out.
7.NS.A.3Eliminate PossibilitiesThe opposite-hands moment is 180/11 minutes after 9, so now is 6 minutes earlier.
Still after 9:00, so the current time is inside the given hour.
7.NS.A.3Convert To AlgebraEach clock hand is a starting angle plus a rate, so a hand-angle question is just a linear equation.
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