# The hands of a clock in some tower are approximately 2m and 1.5m in length. How fast is the distance between the tips of the hands changing at 9:00?

The two vectors, in cartesian coordinates can be written:

The distance from the tips is easily:

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To find how fast the distance between the tips of the clock hands is changing at 9:00, we use the formula for the rate of change of the distance between two moving points. This involves using the Pythagorean theorem and then differentiating with respect to time. Given that the lengths of the clock hands are approximately 2m and 1.5m, respectively, we can proceed with the calculation.

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At 9:00 on a clock, the hour hand and the minute hand are aligned vertically. Let's denote:

( x ) as the distance between the tip of the hour hand and the center of the clock. ( y ) as the distance between the tip of the minute hand and the center of the clock.

Given that the lengths of the hour hand and the minute hand are approximately 2m and 1.5m respectively, at 9:00, ( x = 2 ) meters and ( y = 1.5 ) meters.

The rate of change of ( x ) and ( y ) with respect to time is given by the angular velocity of the hour hand and the minute hand, respectively.

At 9:00, the hour hand is pointing vertically upwards, so its angular velocity is ( \frac{{\pi}}{{6}} ) radians per hour (since the hour hand completes one revolution in 12 hours, and ( 2\pi ) radians is a full revolution).

The minute hand completes one revolution every hour, so its angular velocity is ( \frac{{\pi}}{{30}} ) radians per hour.

To find the rate of change of the distance between the tips of the hands (( z )), we can use the Law of Cosines:

[ z^2 = x^2 + y^2 - 2xy\cos(\theta) ]

Where ( \theta ) is the angle between the hour hand and the minute hand. At 9:00, ( \theta = \frac{{\pi}}{{2}} ) radians.

Differentiating both sides of the equation with respect to time and solving for ( \frac{{dz}}{{dt}} ) (the rate of change of ( z )) yields:

[ \frac{{dz}}{{dt}} = \frac{{x\frac{{dx}}{{dt}} + y\frac{{dy}}{{dt}} - z\sin(\theta)(\frac{{d\theta}}{{dt}})}}{{z}} ]

Substituting the values ( x = 2 ), ( y = 1.5 ), ( \frac{{dx}}{{dt}} = 0 ), ( \frac{{dy}}{{dt}} = \frac{{\pi}}{{30}} ), ( \theta = \frac{{\pi}}{{2}} ), and ( \frac{{d\theta}}{{dt}} = \frac{{\pi}}{{360}} ) into the equation will give the rate of change of ( z ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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