Rate of Change of Clock HandsDate: 08/11/99 at 13:43:19 From: Hoang Nguyen Subject: Related Rates Hi Dr. Math, I am taking calculus and have a problem that I can't find the answer to. I hope that you will help me to find the solution. The problem is: The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o'clock? I need you to show me how to solve this problem step by step, and how to draw a diagram of this problem. Thanks, Hoang Date: 08/11/99 at 14:14:53 From: Doctor Fwg Subject: Re: Related Rates Dear Hoang, Here is a solution for the clock problem. Problem Statement: Find the rate of change (with respect to time) of the distance between the tips of the minute hand and the hour hand of a watch at one o'clock. Note: this drawing shows the time a little after one o'clock so that a more general solution can be worked out. The changing distance is marked "C" on the drawing and is the hypotenuse of the shaded triangle. The length of the minute hand is R and the length of the hour hand is r, where R > r. The hypotenuse ("C") of the shaded triangle may be expressed as a function of its base (X) and height (Y), where: X = r Sin(Theta) - R Sin(Phi) and Y = R Cos(Phi) - r Cos(Theta) Since the shaded triangle is a right triangle, one may write: C^2 = X^2 + Y^2 or C^2 = [r Sin(Theta)-R Sin(Phi)]^2 + [R Cos(Phi)-r Cos(Theta)]^2 or C^2 = r^2 + R^2 - 2rR[Sin(Theta)Sin(Phi) + Cos(Theta)Cos(Phi)] So, using the Theorem of Pythagoras, C may be written as: C = {r^2 + R^2 - 2rR[Sin(Theta)Sin(Phi) + Cos(Theta)Cos(Phi)]}^(1/2) Using the expression above for C, take the full derivative of C with respect to time [i.e., (dC/dt)] to find the rate of change (WRT time) of the distance between the tips of the minute hand and the hour hand of a watch at one o'clock. Remember the following, at one o'clock: Theta = 0 Radians Phi = (2 Pi r)/(12 r) Radians = (Pi/6) Radians d(Theta)/dt = (2 Pi) Radians/12 hr d(Phi)/dt = (2 Pi) Radians/min and the final solution will contain d(Theta)/dt and d(Phi)/dt terms. I hope this is what you were after. - Doctor Fwg, The Math Forum http://mathforum.org/dr.math/ |
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