On a certain clock the minute hand is 10cm long and the hour hand is 7cm long. How fast is the distance between tips of the hands changing if the angle between the clock is 120 degrees?
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a = length of hour hand
b = length of minute hand
c = distance between the tips of the hands
x = angle between the hands
c^2 = a^2+b^2-2abcosx
dc^2/dt = 0+0-2(7)(10)(dcosx/dt)
2c(dc/dt) = -140(-sinx)(dx/dt)
dc/dt = 70sinx/c(dx/dt)
Here's the part I'm not sure about. The hour hand is rotating clockwise at pi/360min and the minute hand is rotating clockwise at pi/30min. When the angle is 2pi/3 (120 degrees), the distance between the hands could either be increasing or decreasing. So dx/dt = pi/30-pi/360 or pi/360-pi/30 = +/-11pi/360. Is that correct?
Also, I get c = sqrt(219) so [dc/dt | x=2pi/3] = [70sin(2pi/3)/sqrt(219)](+/-11pi/360) = +/- 0.39cm/min.
I don't have the answer key but did I mess up anywhere? Any help is appreciated.
***
a = length of hour hand
b = length of minute hand
c = distance between the tips of the hands
x = angle between the hands
c^2 = a^2+b^2-2abcosx
dc^2/dt = 0+0-2(7)(10)(dcosx/dt)
2c(dc/dt) = -140(-sinx)(dx/dt)
dc/dt = 70sinx/c(dx/dt)
Here's the part I'm not sure about. The hour hand is rotating clockwise at pi/360min and the minute hand is rotating clockwise at pi/30min. When the angle is 2pi/3 (120 degrees), the distance between the hands could either be increasing or decreasing. So dx/dt = pi/30-pi/360 or pi/360-pi/30 = +/-11pi/360. Is that correct?
Also, I get c = sqrt(219) so [dc/dt | x=2pi/3] = [70sin(2pi/3)/sqrt(219)](+/-11pi/360) = +/- 0.39cm/min.
I don't have the answer key but did I mess up anywhere? Any help is appreciated.