To find the distance between the tips of the minute and hour hand at 5:40 am, we need to calculate the positions of both hands on the clock face.
Let's start by calculating the position of the hour hand. At 5:40 am, the hour hand would be pointing between the 5 and 6 on the clock dial. We can calculate the angle the hour hand makes with the 12 o'clock position using the formula:
Angle = (hour + (minute / 60)) * (360 / 12)
In this case, the hour is 5 and the minute is 40. Plugging these values into the formula, we get:
Angle = (5 + (40 / 60)) * (360 / 12) = 5.6667 * 30 = 170 degrees
Next, we can calculate the position of the minute hand. At 5:40 am, the minute hand would be pointing exactly at the 8 on the clock dial. The minute hand is directly proportional to the minutes, so we can use the formula:
Angle = minute * (360 / 60)
In this case, the minute is 40. Plugging this value into the formula, we get:
Angle = 40 * (360 / 60) = 40 * 6 = 240 degrees
Now we have the positions of both hands on the clock dial. To find the distance between the tips of the minute and hour hand, we need to calculate the positions of their tips in cartesian coordinates (x, y).
The equation of a circle is given by:
x^2 + y^2 = r^2
where r is the radius of the clock dial (12 inches). Given the angle of the hand, we can calculate the positions (x, y) on the circle using the formulas:
x = r * cos(angle)
y = r * sin(angle)
Let's calculate the positions of the minute hand tip and hour hand tip using these formulas.
For the minute hand:
x_m = 12 * cos(240 degrees) ≈ -6.0 inches
y_m = 12 * sin(240 degrees) ≈ -10.39 inches
For the hour hand:
x_h = 12 * cos(170 degrees) ≈ 2.14 inches
y_h = 12 * sin(170 degrees) ≈ -11.44 inches
Now that we have the positions of both tips, we can calculate the distance between them using the distance formula:
distance = sqrt((x_m - x_h)^2 + (y_m - y_h)^2)
Plugging in the values, we get:
distance = sqrt((-6.0 - 2.14)^2 + (-10.39 - (-11.44))^2)
= sqrt((-8.14)^2 + (1.05)^2)
= sqrt(66.3396 + 1.1025)
≈ sqrt(67.4421)
≈ 8.21 inches
Therefore, the distance between the tips of the minute and hour hand at 5:40 am is approximately 8.21 inches.