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Eliminate the parameter and obtain the standard form of the rectangular equation.
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- Epicycloid If the circle C of Exercise 63 rolls on the outside of the larger circle, the curve traced out by P is called an epicycloid. Find parametric equations for the epicycloid. Hypocycloid A circle C of radius b rolls on the inside of a larger circle of radius a centered at the origin. Let P be a fixed point on the smaller circle, with the initial position at the point (a,0) as shown in the figure. The curve traced out by P is called a hypocycloid. a Show that parametric equations of hypocycloid are x=(ab)cos+bcos(abb) y=(ab)sinbsin(abb) b If a=4b, the hypocycloid is called an asteroid. Show that in this case parametric equations can be reduced to x=acos3y=asin3 Sketch the curve. Eliminate the parameter to obtain an equation for the asteroid in rectangular coordinates.Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.Radiation is focused to an unhealthy area in a patient’s body using a semielliptic reflector, positioned in such a way that the target area is at one focus while the source of radiation is at the other. If the reflector is 100 cm wide and 30 cm high at the center, how far should the radiation source and the target area be from the ends of the reflector?
- Radiation is focused on an unhealthy area in a patient’s body using a semielliptical reflector, positioned in such a way that the target area is at one focus while the source of radiation is at the other. If the reflector is 100 cm wide and 30 cm high at the center, how far should the radiation source and the target area be from the ends of the reflector?The figure shows a period of the curve y = f(x) called “cycloid”, given in parametric form by the equations:x(t) = t − sin ty(t) = 1 − cos tThe curve has a maximum at the point P = (π, 2). Find the equation of the parabola that most closely approximates the cycloid near the point P.Radiation is focused to an unhealthy area in a patient’s body using a semi-elliptic reflector, positioned in such a way that the target area is at one focus while the source of radiation is at the other. If the reflector is 100 cm wide and 30 cm high at the center, how far should the radiation source and the target area be from the end of the reflector?
- radiation is focused to an unhealthy area in a patient’s body using a semi elliptical reflector, positioned in such a way that the target area is at one focus while the source of radiation is at the other. if the reflector is 100 cm wide and 30 cm high at the center, how far should the radiation source and the target area be from the ends of the reflector?Consider the parametric equations x = a cos3 t and y = a sin3 t with 0 ≤ t ≤ π. Find the surface area of the solid obtained by rotating the region about the x-axis.A torus of radius 2 (and cross-sectional radius 1) can be represented parametrically by the function r:D→R3:r(θ,ϕ)=((2+cosϕ)cosθ,(2+cosϕ)sinθ,sinϕ)where D is the rectangle given by 0≤θ≤2π, 0≤ϕ≤2π.The surface area of the torus is
- A. Find the arc length of the curve c(t) = (x(t),y(t)) = (sin(3t), cos(3t)) for 0 les or equal to t and less or equal to π B. Express the Cartesian coordinate (2, 3) in polar coordinates in at least three different ways C. Consider the four petaled rose r = sin(2θ). Find the area of one leaf, then prove that the total area of the rose is equal toone-half the area of the circumscribed circle.Without using a graphing utility, show that the parametric curve r(t) = (3t cos(3t), 3t sin(3t), 3t) lies on the surface with equation x2 + y2 − z2 = 0 and sketch the curve.A simplified model of the Earth-Mars system assumes that the orbits of Earth and Mars are circular with radii of 2 and 3 respectively, and that Earth completes a complete orbit in one year while Mars takes two years. The position of Mars as seen from Earth is given by the parametric equations x= (3−4 cosπt) cosπt+2,y= (3−4 cosπt) sinπt. a. Graph the parametric equations for 0≤t≤2. b. Letting r= 3−4 cosπt, explain why the path of Mars as seen from Earth is a limacon