Describe the distance of point P from the far left or right of the center of the wheel as a function of the angle of the rotation. Assume that the wheel is rotating counterclockwise and the angle of rotation,theta, is the angle between the horizontal ray through the center of the wheel directed to the right and the radius CP. Find x. image:
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a. Describe the horizontal distance from the point P on the wheel to the stirrer blade as a function of the angle of rotation. Find z.
b. How far is the stirrer blade from the center of the wheel for each angle of rotation? Write a function for the distance in terms of the angle.
To describe the distance of point P from the far left or right of the center of the wheel as a function of the angle of rotation, we can start by analyzing the given image.
From the image, we see that the point P is on the circumference of a circle, which represents the wheel. The center of the wheel is denoted as C. We are also given the angle of rotation, θ, which is defined as the angle between the horizontal ray through the center of the wheel directed to the right and the radius CP.
a. To find the horizontal distance from point P on the wheel to the stirrer blade (z), we need to consider the right triangle formed by the radius CP, the horizontal ray, and the horizontal distance (z).
Using trigonometry, we can determine that z = CP * cos(θ). In this case, CP is the radius of the wheel.
b. To determine how far the stirrer blade is from the center of the wheel for each angle of rotation, we can look at the right triangle formed by the radius CP, the vertical distance (y), and the distance from the center of the wheel to the stirrer blade (x).
Using trigonometry again, we can determine that x = CP * sin(θ), since sin(θ) represents the ratio of the opposite side (x) to the hypotenuse (CP).
Therefore, the function for the distance of the stirrer blade from the center of the wheel in terms of the angle of rotation (θ) is:
x = CP * sin(θ)
Note: CP represents the length of the radius from the center of the wheel to point P on the circumference.