Question

The tip of the minute hand on a watch is 12 mm from the center, whereas the tip of the hour hand is 6 mm from the center. How fast is the distance between the tips of the two hands changing at 3 p.m.?

a) 0.2 mm/min
b) 0.3 mm/min
c) 0.4 mm/min
d) 0.5 mm/min

Answer 1

The distance between the tips of the hour and minute hands can be found using the Pythagorean theorem. At 3 p.m., the rate of change of this distance is approximately 0.2 mm/min.

Step-by-step explanation:

The distance between the tips of the hour and minute hands of a watch can be found using the Pythagorean theorem. Let's call this distance 'd'.

d = sqrt((12 - 6)^2 + (0)^2) = 6 mm.

To find how fast this distance is changing at 3 p.m., we need to find the rates of change of the hour and minute hands at that time.

The minute hand completes one revolution in 60 minutes, so its angular speed is 2*pi/60 radians per minute. The hour hand completes one revolution in 12 hours, or 720 minutes, so its angular speed is 2*pi/720 radians per minute.

The rate of change of 'd' can be found using the formula:

ddt = sqrt((d/dt(hour hand))^2 + (d/dt(minute hand))^2).

Substituting the given values, ddt = sqrt((2*pi/720*6)^2 + (2*pi/60*6)^2) = 2*pi/10 = 0.2*pi mm/min.

The approximate value of pi is 3.14, so ddt = 0.2*3.14 = 0.628 mm/min.

So the correct answer is a) 0.2 mm/min.