SOLUTION: What is the distance between the tips of the minute hand and the hour hand of a clock at 1:35pm where the length of the minute hand is 14cm and the length of the hour hand is 9cm

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Question 1207171: What is the distance between the tips of the minute hand and the hour hand of a clock at 1:35pm where the length of the minute hand is 14cm and the length of the hour hand is 9cm
Answer by ikleyn(52788) About Me  (Show Source):
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What is the distance between the tips of the minute hand and the hour hand
of a clock at 1:35pm where the length of the minute hand is 14cm
and the length of the hour hand is 9cm
~~~~~~~~~~~~~~~~~~~ 

At 1:35 pm, the angle between the minute hand and vertical direction up is  

    %2835%2F60%29%2A2pi = %2835%2F30%29pi = %287%2F6%29pi  radians

(since the minute hand makes a full rotation in 60 minutes).


The angle between the hour hand and vertical direction up is

     %281%2F12%29%2A2pi + %2835%2F%2812%2A60%29%29%2A2pi = %281%2F6%29pi + %287%2F72%29pi = %2819%2F72%29pi.

(since the hour hand makes a full rotation in 12 hours = 12*60 minutes).


The difference between these angles is

    %287%2F6%29pi - %2819%2F72%29pi = %2884%2F72%29pi - %2819%2F72%29pi = %2865%2F72%29pi.


Thus the angle between the two hands at 1:35 pm is  %2865%2F72%29pi  radians.

It is more than the right angle pi%2F2 = %2836%2F72%29pi, but less than the straight angle of pi = %2872%2F72%29pi.



Now, we have an obtuse triangle with the sides of 14 cm and 9 cm and the angle of %2865%2F72%29pi  radians between them.


To find the distance between the tips of the hands, apply the cosine law equation

    d = sqrt%2814%5E2%2B9%5E2-2%2A14%2A9%2Acos%28%2865%2F72%29%2A3.14159265%29%29 = 22.745 cm.


ANSWER.  The distance between the tips is about  22.745 cm.

Solved.