Answer :
By differentiating the law of cosines equation the hands that are shifting at one o'clock are separated by 2 millimeters.
By using the law of cosines and the angles of the base and perpendicular, the minute hand is 4 mm and the hour hand is 3 mm.
h² = b² + p² - 2bpcosh
h² = 4² + 3² - 2 × 4 × 3 × cosh
h² = 16 + 9 - (24 × cosh)
h² = 25 - 24cosh
Distinguish both sides in regards to t.
2h (dh/dt) = 24sinh
The cosh angle is 30 degrees at one o'clock.
sin(30) = (1 ÷ 2) dh/dt
Sin 30 = 2pi - pi/6 = 11pi/6
Sin 30's value is converted to h² = 25 - 24cosh,
h² = 25 - 24cos30°
h² = 25 - 20.784
h = 2.0532 = 2.0
So the distance between the hands is closing at a rate of 2 mm per hour.
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