The hands of a clock in some towers are 4.5m and 2m long. How fast does the distance between the tips of the hands change at 9:00 a.m.? (Tip: Use the law of cosines) The distance between the extremities of the hands changes at a rate of _______ m/h at 9:00 a.m.?
I tried several times, mathlab shows the correct answer as a two digit integer with the following decimal places rounded to the tenth place.
After following several different avenues of approach, I get a three-digit number.
Any help is welcome,
Thanks, I followed this example, where am I missing something or doing it wrong?
If we take y the angle between the two hands and x the distance between the two points, then, by the law of cosines, we have:
x^2 = 5^2 + 1.5^2 – 2*5*1.5cos(y)
x^2 = 27.5 – 15cos(y)
Take the derivative of both sides with respect to t, time.
2x*dx/dt = 15sin(y)*dy/dt
Since it is 9:00, the angle between the two hands must be y = π/2. And since there is a right triangle, x = √(5^2 + 1.5^2) = √27.5. To find dy/dt, consider the fact that the hour hand turns around 2π in one hour and the minute hand turns around 2π in 1/60 hour, so we have dy/dt = 2π – 2π/ (1/60) = -118π. Plug it all in:
2(√27.5)*dx/dt = 15sin(π/2)*(-118π)
2(√27.5)*dx/dt = 15(1)*(-118π)
2(√27.5)*dx/dt = -1770π
(√27.5)*dx/dt = -885π
dx/dt = -885π/√27.5 ≈ -530.184 or Ø the angle between them
the angular speed of the minute hand = 2π/60 rad/min = π/30 rad/min
the angular speed of the hour hand = 2π/(12(60)) or π/720 rad/min
then, therefore dØ/dt = (π/30 – π/720) rad/min
dØ/dt = 23π/720 rad/min
Let dm be the distance between the tips of the hands
d^2 = 4.5^2 + 2^2 – 2(2)(4.5)cosØ
d^2 = 24.25 – 18cosØ
differentiate with respect to t
2d dd/dt = 0 + 18sine dØ/dt
Now at 9:00, the angle Ø = 90, and
the angle Ø between the minute hand and the hour hand increases to 23π/720 rad/min
d^2 = 24.25 – 18cos9°0 = 24.25 – 0
d = √24.25
dd/dt = 18 sin90° (23π/720) / 2√24.25
= 0.1834m/min
Where
11.005 m/h
check my arithmetic. Thank you for your reply,
I tried the method mentioned above three times and was incorrect each time, I checked all my work to match the method above.
The correct answer is always just .3 below the answer
All your arithmetic is right too, so that’s not it.
Would it be do/dt?
where is the angle between the hands of the clock?
Please any help is greatly appreciated
I’ve been stuck on this one for days