A man is standing 12 m away from 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... A man is standing 12 m away from 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?
Understand the Problem
The question is asking us to determine the distance between the man's head and the spinning wheel as it rotates. This involves calculating the height of the wheel and the position of the man's head relative to the wheel's diameter and height. We need to find the closest distance from the man's head to the edge of the spinning wheel.
Answer
The closest distance from the man's head to the edge of the wheel is calculated as $H - R$.
Answer for screen readers
The final answer will depend on the man's height and the wheel's radius. For example, if $H = 6$ ft (6 feet) and $R = 2$ ft (2 feet), then the closest distance from the man’s head to the edge of the wheel is:
$$ d_{closest} = H - R = 6 - 2 = 4 \text{ ft} $$
Steps to Solve
- Identify the relevant measurements
First, we need to identify the height of the man and the radius of the wheel. Let’s say the man's height is $H$ and the radius of the wheel is $R$.
- Determine the position of the man's head
The position of the man's head is at height $H$. If the wheel is sitting flat on the ground, then the top of the wheel is at height $R$ above the ground.
- Calculate the distance from the man's head to the top of the wheel
We find the distance from the man's head to the center of the wheel, and then from the center to the top for height considerations. The distance is given by the difference in height: $$ d = (H + R) - H = R $$
- Calculate the distance to the edge of the wheel
The distance from the man's head to the edge of the wheel (which may be directly perpendicular to the man's position) is: $$ d_{edge} = R - \text{(the distance of man from the center of the wheel in horizontal direction)} $$
- Find the closest point
Finally, if the man's position is directly aligned with the center of the wheel, the closest distance will simply be equal to his height minus the radius of the wheel. If he is standing beside it, the distance will slightly increase by his lateral displacement.
The final answer will depend on the man's height and the wheel's radius. For example, if $H = 6$ ft (6 feet) and $R = 2$ ft (2 feet), then the closest distance from the man’s head to the edge of the wheel is:
$$ d_{closest} = H - R = 6 - 2 = 4 \text{ ft} $$
More Information
In this problem, we are looking at the geometric relationship between a person's height and the height of an object. Understanding how to calculate distances with points is crucial in geometry.
Tips
- Not aligning the height measurements correctly with the distances.
- Forgetting to account for where the man is standing relative to the wheel's position.
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