A man is standing 14 m away from the centre of a spinning wheel that is 4 m off the ground, spins at 4 rotations a second and has a diameter of 3 m. If the man is 1.5 m tall, how close does his head get from any part of the wheel?
You leave a lot unsaid, but if the wheel is on a vertical axis, then the distance from the man's head to the closest part of the wheel is
√(14-1.5)^2 + (4-1.5)^2) = 12.7 m
the distance to the farthest spot on the wheel would be
√((14+1.5)^2 + (4-1.5)^2) = 15.7 m
the speed of the wheel is irrelevant.
To find out how close the man's head gets to any part of the spinning wheel, we need to calculate the maximum vertical distance between the man's head and the wheel.
First, let's find the maximum vertical distance by considering the man's head and the highest point of the wheel. The highest point of the wheel is its center plus half of its diameter, which is 4/2 = 2 meters above the ground.
Next, we need to calculate the distance between the man's head and the highest point of the wheel. The man's height is 1.5 meters, and the highest point of the wheel is 2 meters above the ground. Therefore, the vertical distance between the man's head and the highest point of the wheel is 2 - 1.5 = 0.5 meters.
Now, let's find the horizontal distance between the man and the center of the spinning wheel. The man is standing 14 meters away from the center of the wheel.
To calculate the shortest distance between the man's head and any part of the wheel, we need to consider the radius of the wheel. The radius of the wheel is half of its diameter, which is 3/2 = 1.5 meters.
Using the Pythagorean theorem, we can calculate the shortest distance between the man's head and any part of the wheel:
Shortest distance = √((horizontal distance)^2 + (vertical distance)^2)
= √((14)^2 + (0.5)^2)
≈ √(196 + 0.25)
≈ √196.25
≈ 14.01 meters
Therefore, the man's head gets as close as approximately 14.01 meters from any part of the wheel.
To find out how close the man's head gets to any part of the spinning wheel, we need to determine the maximum height of the wheel.
The spinning wheel has a diameter of 3 m, so the radius is half of that, which is 1.5 m.
Since the wheel is 4 m off the ground, the total height of the wheel is the sum of the radius and the distance off the ground, which is 1.5 m + 4 m = 5.5 m.
Next, we need to find the distance between the man and the wheel at its closest point. We can use the Pythagorean theorem to do this.
The man is 14 m away from the center of the wheel, and the height of the wheel is 5.5 m. So, we have a right triangle with a base of 14 m, a height of 5.5 m, and we need to find the hypotenuse, which represents the distance between the man and the closest point on the wheel.
Using the Pythagorean theorem, we can calculate the hypotenuse.
Hypotenuse^2 = Base^2 + Height^2
Hypotenuse^2 = 14^2 + 5.5^2
Hypotenuse^2 = 196 + 30.25
Hypotenuse^2 = 226.25
Taking the square root of both sides gives us the hypotenuse.
Hypotenuse = √226.25 = 15.04
Therefore, the man's head gets as close as approximately 15.04 meters to any part of the spinning wheel.