We are tasked with determining how fast the distance between the tips of the hands of a clock is changing at 9:00, where the lengths of the hands are 4 meters (hour hand) and 1.5 meters (minute hand). The hint suggests using the Law of Cosines.

Step-by-Step Solution:

1. Law of Cosines Setup: Let the lengths of the clock hands be:

   • $L_1 = 4 \, \text{m}$ (hour hand)
   • $L_2 = 1.5 \, \text{m}$ (minute hand)

   The angle $\theta$ between the hands at 9:00 is 90° (since the hour hand points at 9 and the minute hand points at 12, forming a right angle). In radians, this is: $\theta = \frac{\pi}{2} \, \text{radians}.$

   The distance $d$ between the tips of the hands can be calculated using the Law of Cosines: $d^2 = L_1^2 + L_2^2 - 2L_1 L_2 \cos(\theta).$ Substituting $\cos\left(\frac{\pi}{2}\right) = 0$: $d^2 = 4^2 + 1.5^2 = 16 + 2.25 = 18.25.$ Thus, $d = \sqrt{18.25} \approx 4.27 \, \text{m}.$

2. Rate of Change of the Distance: To find the rate of change of the distance between the tips, we must differentiate $d$ with respect to time $t$. Using the general formula for the rate of change of the distance: $\frac{dd}{dt} = \frac{1}{d} \left( L_1 \frac{dL_1}{dt} + L_2 \frac{dL_2}{dt} - 2 L_1 L_2 \sin(\theta) \frac{d\theta}{dt} \right).$

   • The hands' angular velocities are constant.
   • The minute hand moves at $\frac{d\theta_2}{dt} = 2\pi \, \text{radians per hour} = \frac{\pi}{30} \, \text{radians per minute}$.
   • The hour hand moves much more slowly: $\frac{d\theta_1}{dt} = \frac{\pi}{6} \, \text{radians per hour}$.

However, at 9:00, the angle between the hands is constant (90 degrees), so $\frac{d\theta}{dt} = 0$.

Finally, we plug in these angular velocities and solve for $\frac{dd}{dt}$.

Would you like the full details of the next calculation? Let me know! Here are some related questions to explore:

1. How does the angular velocity of the hour hand compare to that of the minute hand?
2. How would the rate of change in the distance vary at a different time, say 3:00?
3. What happens to the angle between the hands as time progresses from 9:00?
4. What is the maximum possible distance between the tips of the hands?
5. How can we generalize this approach to other times on the clock?

Tip: The Law of Cosines is particularly useful for finding distances when you know the angles and sides of non-right triangles.