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trig: how far does the minute hand of a clock travel?

The minute hand of a clock is 4.5 cm long. How far does the tip of the minute hand travel in 80 minutes?
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Zenoture
Asked:
Zenoture
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4 Solutions
 
sdstuberCommented:
we can not do your homework for you.

it is a violation of the member agreement.  

please post what you have tried and we can help assist
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sdstuberCommented:
here's a hint.  The solution does not require trigonometry
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d-glitchCommented:
You may be familiar with:

     Distance = Rate x Time


How far does the tip of the Hour Hand move in 60 minutes?
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d-glitchCommented:
>> How far does the tip of the Minute Hand move in 60 minutes?
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shadow77Commented:
As the tip of the minute hand travels, it follows a path.  How would you describe that path?  What simple formula do you have for measuring it?  Along what fraction of the path does the tip move in 80 minutes?

Distance = Rate x Time is not the formula you need.  There is no rate in this problem.  You also do not need trig.  The formula you need is more basic and is very common.
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phoffricCommented:
>> Distance = Rate x Time is not the formula you need.
There are different ways to solve a problem. It can be the formula. The rate (cm/hour) can be determined easily.
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phoffricCommented:
@Zenoture,
Some hints were offered, so I think to proceed with this thread, you need to show us what you have so far. If you have questions on the hints, then why not ask about them?
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ZenotureAuthor Commented:
Sorry, Had been away from the computer til now.

From what I got from this problem it went something like this:

t = theta

s = r*t

r = 4.5 cm
t = 80 / 60

2*pi = 60
Because 2*pi counts as a full rotation such as 60 mins = 1 hour with a remainder to 20 minutes left thus making...
t/2*pi = 4/3

Doing some math that gives us... 8*pi/3
Now we just substitute that back into the original equation and...
s = 4.5 (8*pi/3) = 36*pi / 3 = 12*pi

Now I just don't know if i'm correct...
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sdstuberCommented:
yes  12 pi  is exactly right!
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shadow77Commented:
12 pi cm is the correct answer, but a simpler way to view the solution is that:
- the tip travels in a circle;
- in 80 minutes, the tip travels around the circle 80/60, or 4/3 times;
- the circumference of a circle is 2 * pi * r;
- hence the distance traveled is
      (4/3) * 2 * pi * 4.5cm
   = (4/3) * 2 * 4.5 * pi cm
   = (4/3) * 9 * pi cm
   = 12 pi cm
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phoffricCommented:
Another way (Distance = Rate * time):
C=2 pi r
Rate = C/60 (cm/min)
Distance = Rate * time = Rate * 80 = (C/60) * 80 = 4C/3 = 4(2 pi r)/3 = 8/3 * pi * r = 12 pi (cm)
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Another way (Angular speed/displacement):
Angular speed, omega (radians/minute) = (2 pi radians) / (60 minutes) = (pi/30) radians/minute
angular displacement, theta = omega * t = (omega radians/minute) * (80 minutes) = (80 * omega) radians
distance = r * theta = r * (80 * omega) = r * 80 * (pi/30) = 8/3 pi r = 12 pi (cm)
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