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Half Time At 3:00 the hour hand and the minute hand of a clock point in directions that are 90.0
apart. What is the first time after 3:00 that the angle between the two hands has decreased by half to 45.0?

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onyebuchinnaji

The first time after 3:00 that the angle between the two hands has decreased by half to 45.0 is 3:08:11.

Angular velocity of the hour hand

The angular velocity of the hour hand for 12 hours time is calculated as follows;

[tex]\omega_h = \frac{2\pi}{12(3600 \ s)}[/tex]

Angular velocity of the minute hand

The angular velocity of the minute hand for 12 hours time is calculated as follows;

[tex]\omega _m = \frac{2\pi}{3600}[/tex]

Difference in the angular velocity

[tex]\Delta \omega = \frac{2\pi }{3600} - \frac{2\pi }{12(3600)} = \frac{11(2\pi)}{12(3600)} = \frac{22\pi}{12 (3600)}[/tex]

First angle between the two hands in radians

[tex]45^0 = \frac{45}{180} \pi= \frac{\pi}{4}[/tex]

Time after 3:00

The time after 3:00 is calculated as follows;

[tex]t =( \frac{\pi}{4}) /\frac{22\pi}{12(3600)} \\\\t = \frac{\pi }{4} \times \frac{12(3600)}{22\pi} \\\\t = 490.91 \ s\\\\t = 8 \min, \ 10.91 \ s[/tex]

[tex]t \approx 8\min, 11 \ s[/tex]

Thus, the first time after 3:00 that the angle between the two hands has decreased by half to 45.0 is 3:08:11.

Learn more time in angular difference here: https://brainly.com/question/7815

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