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?

Question

Emilia Aguilar · Accepted Answer

It is possible to assume that the two hands are vectors and the center of the clock is the origin of two cartesian axes. The minute hand is a vector whose lenght is #2m,# and its angle is #90°#. The hour hand is a vector whose lenght is #1.5m#, and its angle is #180°#.

The two vectors, in cartesian coordinates can be written:

#vecm(0,2)#

#vech(-1.5,0)#

The distance from the tips is easily:

#d=sqrt(2^2+1.5^2)=sqrt(4+2.25)=sqrt6.25=2.5m#.

Paisley Johnson · Answer

undefined 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.

William Abdalla · Answer

undefined 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 $.

HIX Tutor · Answer

undefined To find out how fast the distance between the tips of the clock hands is changing at 9:00, follow these steps:

1. **Understand the Clock Hands**: The long hand (minute hand) is 2m and the short hand (hour hand) is 1.5m.

2. **Position at 9:00**: At 9:00, the hour hand points at 9 and the minute hand points at 12. They form a right angle (90 degrees).

3. **Use the Pythagorean Theorem**: The distance (d) between the tips of the hands can be calculated as:
   $$
   d^2 = (length \, of \, long \, hand)^2 + (length \, of \, short \, hand)^2
   $$
   $$
   d^2 = (2)^2 + (1.5)^2 = 4 + 2.25 = 6.25
   $$
   So, $d = \sqrt{6.25} = 2.5 \text{ m}$.

4. **Calculate Rate of Change of Distance**:
    - The minute hand moves 360 degrees in 60 minutes (6 degrees per minute).
    - The hour hand moves 360 degrees in 12 hours (0.5 degrees per minute).

5. **Find the angles at 9:00**:
   - The angle of the minute hand from 12:00 is 0 degrees.
   - The angle of the hour hand from 12:00 is 270 degrees (or 90 degrees to the left).

6. **Rate of change of angle**: At 9:00, the angle between them is 90 degrees. The rate of change of this angle is 
   $$ 
   6 - 0.5 = 5.5 \text{ degrees per minute}.
   $$

7. **Convert degrees to radians**: 
   ```
   5.5 degrees = 5.5 * (π/180) = 0.0962 radians per minute.
   ```

8. **Final Calculation**: Use the formula. The distance changes with the angle too. The speed (v) of distance change:
   $$
   v = \frac{d}{dt}\left(\sqrt{(2)^2 + (1.5)^2}\right) \approx 2.5 * 0.0962 = 0.2405 \text{ m/min}.
   $$

So, **the distance between the tips of the hands is changing at about** **0.24 m/min** **at 9:00.**