Question

Please help with this word problem :
On a clock, the minute hand is 4" long and the hour hand is 3" long. How fast is the distance b/t the tips os the hands changing at 9pm. answering in minutes/hour. Thank you!

Answer 1

law of cosines might help ....

Answer 2

Im not sure how to proceed. ? I think it has to do with derivatives?

Answer 3

use law of cosines as @amistre64 said

\[x^2 = 3^2 +4^2 - 2(3)(4) \cos \theta\]
\[x^2 = 25 - 24 \cos \theta\]
take derivative
\[2x \frac{dx}{dt} = 24 \sin \theta \frac{d \theta}{dt}\]
\[\frac{dx}{dt} = \frac{12 \sin \theta}{x} \frac{d \theta}{dt}\]
Now we need to know how fast angle is changing each minute

minute hand ---> moves 60 minutes around circle ---> 2pi = 60--> pi/30 radians/min
hour hand ---> moves 12 hours or 720 minutes around circle --> 2pi=720 --> pi/360
The angle between them is the difference
---> pi/30 - pi/360 = 11pi/360
\[\theta(t) = \frac{11 \pi}{360} t\]
\[\frac{d \theta}{dt} = \frac{11\pi}{360}\]

given value of 9pm = 9 hrs = 540 min
angle simplifies to pi/2
x = 5

plug values in
\[\frac{dx}{dt} = \frac{12 \sin(\pi/2)}{5}*\frac{11\pi}{360}\]
\[\frac{dx}{dt} = \frac{11\pi}{150}\]

Answer 4

What units is that?

Answer 5

I need to come up with inches per hour, and the number I came up with is a bit off from that.

Answer 6

oh i did inches per min .... just multiply by 60 to get in/hr

Answer 7

11 pi /3?

Answer 8

\[\frac{11\pi}{150}*60 = \frac{22\pi}{5}\]

Answer 9

Oh duh, ok, still not what I originally got but I see what I did wrong, thanks!

Answer 10

yw :)