   Chapter 2.8, Problem 50E

Chapter
Section
Textbook Problem

# 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?

To determine

To find:

How fast is the distance between the tips of the hands changing at one o’ clock position?

Explanation

1. Concept:

We can plot the figure from the given situation and then apply the cosine and differentiation to find the rate.

2. Formula:

i. Cosine rule:

L2= a2+b2-abcosθ

ii. ddx(cosx)= -sinx

3. Given:

Length of minute hand = 8 mm, length of hour hand = 4 mm.

4. Calculations:

By cosine rule the distance between the two at given angular distance θ is x2 = 42+82 -248cosθ

Differentiate it with respect to t.

2xdxdt= 64 sinθ

dxdt= 32sinθxdθdt Thus it follows that the distance between the two hands are changing at a rate of dxdt= 32sinθxdθdt

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