<PRE><B>Question 160353</B>
The hour hand of a clock is 4 inches long while the minute hand is 5 inches long. At some time between 12:15 PM and 12:30 PM, the tips of the hands are 8 inches apart. What time is it then? (Answer to the nearest second.)
:
Find the angle (A) between the hands using the law of cosines:
: 
b^2 + c^2 -2(bc)Cos(A) = a^2
Let a=8, b=4, c=5
:
4^2 + 5^2 - 2(4*5)Cos(A) = 8^2
16 + 25 - 2(20)Cos(A) = 64
41 - 40Cos(A) = 64
-40Cos(A) = 64-41
Cos(A) = {{{23/(-40)}}}
A = 125.1 degrees is the angle between the hands
:
Let m = minutes hand position
:
6m = degrees per min
:
Hour hand moves 360/12 = 30 degrees per hr
:
An equation
Hrs degrees + 125.1 degrees = minutes degrees
{{{m/60}}}*30 = 125.1 = 6m
:
{{{m/2}}} + 125.1 = 6m
Multiply equation by 2
m + 250.2 = 12m
:
250.2 = 12m - m
:
250.2 = 11m
m = {{{252.2/11}}}
m = 22.927 min or 22 min + .927(60) = 22 min 56 sec

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