The Epicycloid. When a wheel of radius b rolls around the outside of a circle of radius a, a point on the wheel traces a curve called an epicycloid: x = (a + b)cos(t) - bcos((a+b)/b * t) 0 ? t ? 2b? y = (a + b)sin(t) - bsin((a+b)/b * t) Find the arc length of the epicycloid with a = 5 and b = 1. Show all your steps!
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Given the parametric equations for the epicycloid are: $$x(t) = (a+b)\cos(t) - b\cos\left(\frac{a+b}{b}t\right)$$ $$y(t) = (a+b)\sin(t) - b\sin\left(\frac{a+b}{b}t\right)$$ Show more…
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The curve traced by a point on a circle of radius $b$ as it rolls without slipping on the outside of a fixed circle of radius $a$ is called an epicycloid. Show that it has parametric equations $$ \begin{array}{l} x=(a+b) \cos t-b \cos \frac{a+b}{b} t \\ y=(a+b) \sin t-b \sin \frac{a+b}{b} t \end{array} $$ (See the hint in Problem $61 .$ )
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The arc length L of a curve given parametrically by (x(t), y(t)) for a ≤ t ≤ b is given by the formula L = integral a to b (x'(t))^2 + (y'(t))^2 dt A path of a point on the edge of a rolling circle of radius R is a cycloid, given by x(t) = R(t - sin t), y(t) = R(1 - cos t), where t is the angle (in radians) the circle has rotated. Find the length L of one "arch" of this cycloid, that is, find the distance traveled by a small stone stuck in the tread of a tire of radius R during one revolution of the rolling tire.
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