Question

the clock on the wall at the aops office has an hour hand that's 4 cm long and a minute hand that's 8 cm long. at exactly 2:00, at what rate, in centimeters per minute, is the distance between the tips of the two hands changing?

Answer 1

The rate at which the distance between the tips of the hour and minute hands of a clock is changing can be calculated using the concept of arc length. By differentiating the expression for the arc length of the minute hand with respect to time, we can find the rate of change of distance between the two hands. The formula d(distance)/dt = (2 * π * radius)/360 * (2 * π)/43200 can be used to calculate this rate of change.

Step-by-step explanation:

The distance between the tips of the hour and minute hands of a clock can be found using the concept of arc length. The length of the arc between the two hands can be approximated as the difference in the lengths of the arcs traced by the hour and minute hands. The arc length of the hour hand is given by the formula arc length = (angle/360) * (2 * π * radius), where the angle is the angle between the hour hand and the 12 o'clock position, and the radius is the length of the hour hand. Similarly, the arc length of the minute hand is given by arc length = (angle/360) * (2 * π * radius), where the radius is the length of the minute hand.

Therefore, the rate at which the distance between the tips of the two hands is changing can be found by differentiating the expression for the arc length of the minute hand with respect to time. Since the hour hand is not moving, its rate of change is zero. Therefore, the rate at which the distance between the tips of the two hands is changing is given by d(distance)/dt = d(arc length)/ dt = d(arc length)/ d(angle) * d(angle)/ dt. The derivative of the arc length with respect to the angle can be found using the chain rule and is given by (2 * π * radius)/360. The derivative of the angle with respect to time is the angular velocity, which is constant at 1 revolution per 12 hours or (2 * π)/43200. Finally, substituting these values into the expression for the rate of change of distance, we get d(distance)/dt = (2 * π * radius)/360 * (2 * π)/43200. Plugging in the values of the radius for the hour and minute hands, we can calculate the rate at which the distance between the two hands is changing.