a clock has a dial face of 12 inches radius the minute hand is 9 inches while the hour hand is 6 inches the plane of rotation of the hour hand is 2 inches above the plane of rotation of the minute hand. Find the distance between the tips of the minute and hour hand at 5:40 am.
To find the distance between the tips of the minute and hour hand at 5:40 am, we need to determine the angles at which the hour and minute hands are pointing.
At 5:40 am, the minute hand will be pointing at the 8-minute mark, and the hour hand will be pointing between the 5 and 6-hour marks.
The minute hand moves 360 degrees in 60 minutes, so the angle it makes with the 12 o'clock position is (8/60) * 360 degrees = 48 degrees.
The hour hand moves 360 degrees in 12 hours, so the angle it makes with the 12 o'clock position is (5 + (40/60)) * (360/12) = 190 degrees.
Now, we need to calculate the distance between the tips of the hour and minute hands. Given that the minute hand has a length of 9 inches and the hour hand has a length of 6 inches, we can use the law of cosines to determine this distance.
Let's say the distance between the tips is "d". Using the law of cosines:
d^2 = (9^2) + (6^2) - 2 * 9 * 6 * cos(190 - 48)
Let's calculate that:
d^2 = 81 + 36 - 108 * cos(142)
We can use a calculator to find the value of cos(142):
cos(142) ≈ -0.766
Substituting the value into the equation:
d^2 = 81 + 36 - 108 * (-0.766)
d^2 ≈ 81 + 36 + 82.848
d^2 ≈ 199.848
Taking the square root of both sides:
d ≈ sqrt(199.848)
d ≈ 14.14
Therefore, the distance between the tips of the minute and hour hand at 5:40 am is approximately 14.14 inches.
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.
let the plane of rotation of the minute hand be z=0. So, the location of the tip of the minute hand is
(9cosθ,9sinθ,0)
Similarly, the location of the tip of the hour hand is
(6cos θ/12,6sin θ/12,2)
at 5:40 am, θ = 2π/3, so θ/12 = π/18
So, the distance between the tips is
d^2 = (9cos 2π/3 - 6cos π/18)^2
+ (9sin 2π/3 - 6sin π/18)^2
+ 4
now just evaluate for d
oops. Forgot all those hours
θ = 2π*5 + 2π/3 = 10π + 2π/3 = 32π/3
For the minute hand, that's the same value as 2π/3
but, θ/12 is now 32π/36 = 8π/9
which affects its trig function values immensely.