Question 1179575: Now explore the position of the redsaw tooth in reference to an imaginary verticalaxis of symmetry of the circular blade. The red toothis initially one foot to the right of the dottedline.How far to the right of this axis is the tooth after37 seconds? After 237 seconds? After t seconds? Drawby hand a graph that shows how the displacement pof the red tooth with respect to the vertical axisis afunction of the elapsed time t.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! It seems like you're describing a scenario with a circular saw blade and a red tooth on the blade. Here's how we can analyze the motion of the red tooth:
**Understanding the Motion**
* **Circular Motion:** The red tooth is moving in a circle around the center of the saw blade.
* **Periodic Motion:** The tooth repeatedly returns to the same position after each full rotation of the blade.
* **Displacement and the Vertical Axis:** We're interested in how far the tooth is to the right or left of the vertical axis of symmetry. This is its horizontal displacement.
**Assumptions**
To make this problem solvable, let's assume:
* **Constant Speed:** The saw blade is rotating at a constant speed.
* **Clockwise Rotation:** The blade is rotating in a clockwise direction.
**Analysis**
* **Initial Position:** The tooth starts 1 foot to the right of the vertical axis.
* **After 37 seconds:** Without knowing the speed of rotation, we can't determine the exact position. However, we know the tooth will be somewhere along the circle, and its horizontal displacement will depend on how many rotations it has completed.
* **After 237 seconds:** Similar to the previous case, the position depends on the rotation speed.
* **After t seconds:** We need to know the time it takes for one complete rotation (the period) to determine the position after 't' seconds.
**Graphing the Displacement**
Since we don't have the exact rotation speed, we can create a general graph to represent the motion:
1. **Axes:**
* Horizontal axis (x-axis): Time (t) in seconds
* Vertical axis (y-axis): Horizontal displacement (p) in feet, with positive values to the right and negative values to the left.
2. **Shape of the Graph:**
* The graph will be a sine wave or a cosine wave since the motion is periodic.
* The amplitude of the wave will be 1 foot (the maximum distance from the vertical axis).
* The period of the wave will depend on the rotation speed.
**Example Graph (assuming a period of 60 seconds)**
[asy]
unitsize(2 cm);
real func (real x) {
return(Cos(pi*x/30));
}
draw(graph(func,-65,65));
draw((-65,0)--(65,0));
draw((0,-1.2)--(0,1.2));
label("$t$", (65,0), E);
label("$p$", (0,1.2), N);
dot("$1$", (0,1), NE);
[/asy]
**Key Points on the Graph:**
* The graph starts at a maximum value of 1 (initial position).
* It crosses the x-axis at t = 15 seconds and t = 45 seconds (when the tooth is on the vertical axis).
* It reaches a minimum value of -1 at t = 30 seconds (when the tooth is farthest to the left).
**To get a more precise graph and answer the questions about the position at 37 seconds and 237 seconds, we need to know the rotation speed of the saw blade.**
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
In this post, key necessary input data are missed, due to the author's negligence.
So, the problem is DEFECTIVE: it is mutilated and cannot be solved.
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