Watch Hands
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Emporio Armani Men's Classic Three-hand watch #AR0405 List Price: $175.00 Sale Price: $138.00  |
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Classic - Three-hand Japanese Quartz movement displaying polished silver-tone hour, minute and sweep seconds hands, Applied and polished silver-tone 'EA Eagle' marks 12 hour, Applied and polished silver-tone baton hour markers, White indices reveal outer minute track, Analog display, Polished silver-tone stainless steel rectangular case etched with Armani logo name at case side, Notched and polished crown, Stainless steel caseback, 50 meters/165 feet water resistant |
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Calculus Related Rates, Watch Hands?
The minute hand on a watch is 6 mm long and the hour hand is 3 mm long. How fast is the distance between the tips of the hands changing at one o'clock?
D^2 = 6^2 + 3^2 - 2*6*3cos x
= 36 + 9 - 36cos x
= 45 - 36cos x
differentiated,
2D dD = 36sin x dx
dD/dx = 36sin x / 2D = 18sin x / D
at 1 o'clock, x = π/6
dx/dt = 30*/hour (of hour hand) - 360*/hour (of minute hand)
= 0.5*/minute - 6*/minute
= -5.5*/minute
dD/dt = dD/dx * dx/dt
= [18sin 30* / D] * [-5.5]
= [18(0.5) / √(45 - 18√3)] * -5.5
= -13.31382 mm/minute
negative dD/dt means the distance is closing (becoming closer).