Related Rates: The minute hand on a watch is 9 mm long and the hour hand is 6 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 rate at which the distance between the tips of the hour and minute hands is changing at one o'clock, we can use the concept of related rates.
Let's assume that the center of the clock is the origin of a coordinate system. The position of the minute hand can be described by a point moving in a circle centered at the origin with a radius of 9 mm (the length of the minute hand), while the position of the hour hand can be described by a point moving in a smaller circle centered at the origin with a radius of 6 mm (the length of the hour hand).
At one o'clock, the minute hand will be pointing directly at the 12, while the hour hand will be pointing at the 1. Let's denote the position of the minute hand as (x1, y1) and the position of the hour hand as (x2, y2).
Since the minute hand is rotating at a constant rate of 360 degrees per hour, we can express the position of the minute hand as:
x1 = 9 cos(t)
y1 = 9 sin(t)
where t is the time in hours. Since we are interested in one o'clock, t = 1.
Similarly, the hour hand is rotating at a constant rate of 30 degrees per hour, so we can express the position of the hour hand as:
x2 = 6 cos(30t)
y2 = 6 sin(30t)
We are interested in finding the rate at which the distance between the tips of the hands is changing, which can be represented by the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
To find the rate at which d is changing, we need to differentiate d with respect to time. Let's call this rate of change of d with respect to time as dd/dt.
dd/dt = d/dt (sqrt((x2 - x1)^2 + (y2 - y1)^2))
Now, let's differentiate each component separately. The chain rule will be used in each step.
Using the formula for distance between two points, we have:
d^2 = (x2 - x1)^2 + (y2 - y1)^2
Differentiating both sides with respect to t:
2d * dd/dt = 2(x2 - x1) * (dx2/dt - dx1/dt) + 2(y2 - y1) * (dy2/dt - dy1/dt)
Simplifying the equation:
d * dd/dt = (x2 - x1) * (dx2/dt - dx1/dt) + (y2 - y1) * (dy2/dt - dy1/dt)
Now, let's substitute the expressions we found earlier for x1, y1, x2, and y2:
d * dd/dt = (6 cos(30t) - 9 cos(t)) * (-6 * 30 sin(30t) - 9 sin(t)) + (6 sin(30t) - 9 sin(t)) * (6 * 30 cos(30t) - 9 cos(t))
Evaluating this expression at t = 1:
d * dd/dt = (6 cos(30) - 9 cos(1)) * (-6 * 30 sin(30) - 9 sin(1)) + (6 sin(30) - 9 sin(1)) * (6 * 30 cos(30) - 9 cos(1))
Calculating the value of dd/dt will give us the rate at which the distance between the tips of the hands is changing at one o'clock. Round the answer to one decimal place.
To determine how fast the distance between the tips of the hands is changing, we can use the concept of related rates. Let's denote the distance between the tips of the hour and minute hands as "d."
At one o'clock, the minute hand is at the 12 o'clock position, pointing upward, and the hour hand is at the 1 o'clock position, slightly to the right. The minute hand moves 360 degrees in 60 minutes, so its angular velocity is given by:
angular velocity of the minute hand = (360 degrees) / (60 minutes) = 6 degrees per minute.
The hour hand moves 360 degrees in 12 hours or 720 minutes, so its angular velocity is given by:
angular velocity of the hour hand = (360 degrees) / (720 minutes) = 0.5 degrees per minute.
To find the rate of change of the distance between the tips of the hands (ddt), we can use the following formula:
ddt = sqrt((dxdt)^2 + (dydt)^2),
where dxdt and dydt are the rates of change of the x-coordinate and y-coordinate of the minute hand, respectively.
To find dxdt, the rate of change of the x-coordinate of the minute hand, we can use the following formula:
dxdt = length of minute hand * angular velocity of the minute hand * cosine(theta),
where theta is the angle between the x-axis and the minute hand. At one o'clock, theta is 90 degrees.
Substituting the given values, we have:
dxdt = 9 mm * (6 degrees per minute) * cos(90 degrees) = 0 mm per minute.
Since the minute hand only moves vertically, its x-coordinate remains constant, resulting in dxdt = 0.
To find dydt, the rate of change of the y-coordinate of the minute hand, we can use the following formula:
dydt = length of minute hand * angular velocity of the minute hand * sine(theta),
where again, theta is 90 degrees.
Substituting the given values, we have:
dydt = 9 mm * (6 degrees per minute) * sin(90 degrees) = 54 mm per minute.
Note that sine(90 degrees) = 1.
Now, we can calculate ddt, the rate of change of the distance between the tips of the hands:
ddt = sqrt((dxdt)^2 + (dydt)^2) = sqrt((0 mm per minute)^2 + (54 mm per minute)^2) = sqrt(0 + 2916) = sqrt(2916) = 54 mm per minute.
Therefore, at one o'clock, the distance between the tips of the hands is changing at a rate of 54 mm per minute.
If the hands are separated by an angle θ, then using the law of cosines, the distance z between the tips is
z^2 = 6^2 + 9^2 - 2(6)(9) cosθ
The hour hand rotates at a speed of 2π/1440 radians/min, and the minute hand moves at 2π/60 radians/min.
So, figure θ at 1:00 and you know that dθ/dt = -23π/720
Now you can find z at 1:00 and you can find dz/dt using
2z dz/dt = 2(6)(9) sinθ dθ/dt