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Math Help - Help! Angles and Clocks

  1. #1
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    Help! Angles and Clocks

    How would I go about figuring this out?

    Mario was studying angles and wondered if during his math class the minute and hour hands of the clock formed a right angle. If his class meets from 6:00 P.M. to 6:50 P.M., is a right angle formed? If so, figure out to the nearest second when the hands form the right angle.
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    MHF Contributor MarkFL's Avatar
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    Re: Help! Angles and Clocks

    What are the initial positions of the two hands, and at what angular speed in degrees per hour do the two hands move?
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    Re: Help! Angles and Clocks

    I haven't a clue! Haha. This is all I was given to work with.
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    MHF Contributor MarkFL's Avatar
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    Re: Help! Angles and Clocks

    Using the standard convention for measuring angles on the unit circle, at 6:00 pm, at what angle is the minute hand? The hour hand?
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    Re: Help! Angles and Clocks

    Quote Originally Posted by stallioncane View Post
    I haven't a clue! Haha. This is all I was given to work with.
    FYI: He's giving you hints about how to start.

    -Dan
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    Re: Help! Angles and Clocks

    Well I've got it down to 6:16 but I can't figure out how to do the seconds.
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    MHF Contributor MarkFL's Avatar
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    Re: Help! Angles and Clocks

    That's close to one of the valid times, but without knowing what you did, I can't really help.
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    Re: Help! Angles and Clocks

    Honestly, I'm an early childhood education major. I busted out a Judy Clock and figured it out that way. Is there a particular formula I can use to find the seconds?
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    Re: Help! Angles and Clocks

    Nevermind. Figured it out. Thanks!
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  10. #10
    MHF Contributor MarkFL's Avatar
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    Re: Help! Angles and Clocks

    Are you familiar with angles on the unit circle?
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  11. #11
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    Re: Help! Angles and Clocks

    Hello, stallioncane!

    Here is my approach to Clock-hands Problems.


    Mario was studying angles and wondered if during his math class
    the minute and hour hands of the clock formed a right angle.
    If his class meets from 6:00 P.M. to 6:50 P.M., is a right angle formed?
    If so, figure out to the nearest second when the hands form a right angle.

    You might think that one answer looks like this:
    Code:
                  * * *
              *           *
            *               *
           *                 *
    
          *                   *
          *         *-------- * 3
          *         |         *
                    |
           *        |        *
            *               *
              *           *
                  * * *
                    6
    But this is incorrect.

    By the time the minute hand has moved 90o to the "3",
    . . the hour hand has already moved toward the "7".


    The minute hand moves 360o in 60 minutes.
    It moves 6^o per minute.
    It starts at "12" (0o) and in the next t minutes, it move 6t degrees.
    The position of the minute hand at time t is: . M \:=\:6t degrees.

    The hour hand moves 30o in 60 minutes.
    It move \tfrac{1}{2}^o per minute.
    It starts at "6" (180o) and in the next t minutes, it moves \tfrac{1}{2}t degrees.
    The position on the hour hand at time t is: . H \:=\:180 + \tfrac{1}{2}t degrees.


    There are two possible cases.

    [1] The hour hand is 90o ahead of the minute hand: . H \:=\:M + 90

    . . . 180 + \tfrac{1}{2}t \:=\:6t + 90 \quad\Rightarrow\quad \tfrac{11}{2}t \:=\:90 \quad\Rightarrow\quad t \:=\:\tfrac{180}{11}

    . . . t \:=\:16\tfrac{4}{11} \:=\:16\text{ minutes, }21.\overline{81}\text{ seconds}

    Time: 16 minutes, 22 seconds after 6 o'clock.


    [2] The minute hand is 90o ahead of the hour hand: . M \:=\:H + 90

    . . . 6t \:=\:180 + \tfrac{1}{2}t 90 \quad\Rightarrow\quad \tfrac{11}{2}t \:=\:270 \quad\Rightarrow\quad t \:=\:\frac{540}{11}

    . . . t \:=\:49\tfrac{1}{11} \:=\:49\text{ minutes, }5.\overline{45}\text{ seconds}

    Time: 49 minutes, 5 seconds after 6 o'clock.
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