SOLUTION: A clock has hour and minute hands 1 and 1.6 inches long respectively. At what rate are the ends of the hands approaching each other when the time is 2 o'clock?

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Question 1183864: A clock has hour and minute hands 1 and 1.6 inches long respectively. At what rate are the ends of the hands approaching each other when the time is 2 o'clock?
Found 2 solutions by robertb, ikleyn:
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!



highlight%28NOTE%29: As a response to @ikleyn, the problem is basically one on RELATED RATES in Calculus, which entails relating the distance
between the tips of the minute and hour hands with the rate of change of angle between them. So this is an EASY problem.

I don't really need to justify my solution, as the student has already confirmed/acknowledged that the answer is as what I have given.

So NO CONCEPTUAL ERRORS, just a straightforward application of related rates.

So as far as solutions are concerned, ikleyn's solution is GARBAGE, and needs to be thrown out to the TRASH BIN.

And wash hands afterwards, to prevent spread of virus. :)

P.S.: In ikleyn's solution it put the hour hand as being represented by the unit vector (cos%28pi%2F3%29, sin%28pi%2F3%29) and the minute hand as (1.6%2A%28pi%2F30%29,0).
(She even mad a mistake on the minute hand representation.) VERY, VERY BIG MISTAKES. Remember. at exactly 2 o'clock, the angle between them is pi%2F3, or sixty degrees, SO AT THAT INSTANT, the theta value for the minute hand CANNOT be pi%2F3 at the same time as the hour hand. So her argument is highlight%28FLAWED%29.

Try to read and understand the problem TWICE, THRICE,... And formulate a correct solution in cycles of 8. :)

To ikleyn, just let me know if you need help in Calculus, and I will gladly help you.

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Use the cosine law to relate the lengths of the minute and hour hands with the distance d between their tips and the angle theta, the angle between them.
Then we have



===> d%2A%28dd%2Fdt%29+=+%288%2F5%29%2Asin%28theta%29%2A%28d%28theta%29%2Fdt%29.

At 2 o'clock, the minute and hour hands are exactly pi%2F3 radians (or 60 degrees) apart.
At that instant, the distance between their tips is 7/5 inches. (You can easily verify this!)

We need to get d%28theta%29%2Fdt, which is easily obtained as
pi%2F360+-+pi%2F30+=+-%2811pi%29%2F360.
I will not elaborate on this fact but rather will just leave the reason for your mental exercise, including what the NEGATIVE sign means

===> , to 3 d.p.

Therefore the tips of the minute and hour hands are are getting near each other at the rate of highlight%280.095%29 inch per minute.


Answer by ikleyn(52792) About Me  (Show Source):
You can put this solution on YOUR website!
.

There was a mistake in my solution, which I placed here before.

So I deleted it now.