The minute hand on a watch is 7 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o'clock? (Round your answer to one decimal place.)
To find the speed at which the distance between the tips of the hands is changing at one o'clock, we can use the concept of derivatives.
Let's consider the position of the hour hand and the minute hand at one o'clock. At one o'clock, the hour hand is pointing directly at the 1 on the watch, while the minute hand is pointing directly at the 12. We can think of this as an angle formed between the two hands.
To start, let's assume that each hand moves at a constant speed. The minute hand completes a full rotation every 60 minutes or 360 degrees, while the hour hand completes a full rotation every 12 hours, or 720 minutes, which is equal to 30 degrees per hour.
Now, let's find an expression for the distance between the tips of the hands. We can think of this as the hypotenuse of a right triangle, with the hour hand as one side and the minute hand as the other side. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using this, we have:
distance^2 = (length of hour hand)^2 + (length of minute hand)^2
distance^2 = 4^2 + 7^2
distance^2 = 16 + 49
distance^2 = 65
distance = √65 ≈ 8.06 mm
Now, let's differentiate both sides of this equation with respect to time (t), assuming that the distance is changing with respect to time.
2 * distance * (rate of change of distance) = 2 * (length of hour hand) * (rate of change of hour hand) + 2 * (length of minute hand) * (rate of change of minute hand)
(rate of change of distance) = [(length of hour hand) * (rate of change of hour hand) + (length of minute hand) * (rate of change of minute hand)] / distance
Now, substitute the values we know into this equation:
(rate of change of distance) = [(4 mm) * (30 degrees per hour) + (7 mm) * (360 degrees per hour)] / 8.06 mm
(rate of change of distance) = [120 mm degrees per hour + 2520 mm degrees per hour] / 8.06 mm
(rate of change of distance) = 40 degrees per hour + 312.16 degrees per hour
(rate of change of distance) = 352.16 degrees per hour
Therefore, the speed at which the distance between the tips of the hands is changing at one o'clock is approximately 352.2 degrees per hour.