Below is an arc of the astroid curve æ2/3 +y2/3 = 1, with exactly = 1.125 units of length. Note that the astroid can be parametrized by %3D Y(t) = (cos (t), sin (t)), te (0,27]. In this problem, we will find the coordinates of the point P at the end of the arc. 0.8 0.6 0.4 0.2 -08 -0,6 -04 -0,2 0.2 04 0,6 0.8 (a) Show that the speed of y at time t is | sin(2t)|. (b) At what time te [0, 27] does y trace out exactly = 1.125 arc length units? (Hint: Be careful when integrating absolute values: for example, to evaluate the integral S* | sin(æ)| dx, you have to split it up into two integrals S"(– sin(x)) dx + S* sin(æ) dx. 27
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![Below is an arc of the astroid curve x2/3 +y?/3 = 1, with exactly = 1.125 units of length. Note that
the astroid can be parametrized by
%3D
Y(t) = (cos (t), sin° (t)), te [0, 27].
In this problem, we will find the coordinates of the point P at the end of the arc.
0.8
0.6
0.4
P
0.2
-0.8
-0,6
-04
-0.2
0.2
0.4
0.6
0.8
(a) Show that the speed of y at time t is | sin(2t)|.
(b) At what time t e [0, 27] does y trace out exactly = 1.125 arc length units?
(Hint: Be careful when integrating absolute values: for example, to evaluate the integral
So" | sin(z)| dr, you have to split it up into two integrals So(- sin(x)) dz + S sin(x) dr.
(c) Find the exact Cartesian coordinates of P.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba73b71-75cf-41e8-b53e-fd46ec14d4a7%2F1c03b4b1-d685-401e-934c-a88447792106%2F6c1937_processed.png&w=3840&q=75)
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