A man is standing 12 m away from the middle 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... A man is standing 12 m away from the middle 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?
Understand the Problem
The question is asking to determine the minimum distance from the head of a man to any point of a spinning wheel. The key calculations will involve the height of the wheel, the height of the man, and the distance from the man to the center of the wheel. We need to ascertain how close the man's head comes to the wheel as it spins.
Answer
The minimum distance from the man's head to the edge of the wheel is given by \( D_{min} = \sqrt{(H_w - H_m)^2 + (D_h - R)^2} \).
Answer for screen readers
The minimum distance from the man's head to the edge of the spinning wheel is given by
$$ D_{min} = \sqrt{(H_w - H_m)^2 + (D_h - R)^2} $$
Steps to Solve
- Identify the heights involved
Let ( H_w ) be the height of the wheel and ( H_m ) be the height of the man. The height of the man's head will be determined by how tall he is.
- Determine the vertical distance
The vertical distance from the man's head to the bottom of the wheel can be expressed as:
$$ D_v = H_w - H_m $$
This gives us the height of the man's head relative to the base of the wheel.
- Calculate the horizontal distance
Next, we need to consider how far the man is from the center of the wheel. Let ( D_h ) be the horizontal distance from the man to the wheel's center.
- Calculate the minimum distance to the edge of the wheel
Assuming the wheel has a radius ( R ), the distance from the center of the wheel to the edge is simply ( R ). Therefore, the minimum distance from the man's head to any point on the wheel can be found as follows:
$$ D_{min} = \sqrt{D_v^2 + (D_h - R)^2} $$
- Use actual values to compute
Plug in the actual values for ( H_w ), ( H_m ), ( D_h ), and ( R ) based on the specific scenario to find the minimum distance.
The minimum distance from the man's head to the edge of the spinning wheel is given by
$$ D_{min} = \sqrt{(H_w - H_m)^2 + (D_h - R)^2} $$
More Information
This formula provides a geometric interpretation of the problem by calculating the straight-line distance between two points: the man's head and the edge of the spinning wheel. Understanding the relationship between these distances will help solve similar problems in physics and geometry.
Tips
- Forgetting to consider both vertical and horizontal components of the distance.
- Miscalculating the radius of the wheel or the heights involved.
- Not applying the Pythagorean theorem properly when combining the distance components.
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