Question

Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. x= -7 cos 2t, y= -7 sin 2t, 0 leq t leq pi

Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. \(x=\ -7\ \cos\ 2t,\ y=\ -7\ \sin\ 2t,\ 0\ \leq\ t\ \leq\ \pi\)

Answers (1)

2021-02-22

Step 1 Given: The following parametric equations, \(x=\ -7\ \cos(2t),\ y=\ -7\ \sin(2t),\ 0\ \leq\ t\ \leq\ \pi.\)

Step 2 a) To eliminate the parameter to obtain an equation in x and y. We have, The following parametric equations, (1)\(x =\ −7 \cos(2t)\)
\(y =\ −7 \sin(2t)\)(2) Adding equations (1) and (2), we get \(x^{2}\ +\ y^{2}=(-7\ \cos(2t))^{2}\ +\ (-7\ \sin(2t))^{2}\)
\(=49\ \cos^{2}(2t)\ +\ 49\ \sin^{2}(2t)\)
\(=49(\cos^{2}(2t)\ +\ \sin^{2}(2t))\)
\(=49(1)\ \because \text{By using trignometric identity}\)
\(=49\) An equation in x and y is \(x^{2}\ +\ y^{2} = 49.\) Step 3 b) To describe the curve and indicate the positive orientation. We have, \(x^{2}\ +\ y^{2} = (7)^{2}.\) The curve represents a circle at the origin, \((0,\ 0)\ \text{and radius,}\ r = 7.\) To indicate the positive orientation, we will use the following parametric equations, \(x =\ −7\ \cos(2t),\ y =\ −7\ \sin(2t),\ 0\ \leq\ t\ \leq\ \pi.\) The interval given is \(0\ \leq\ t\ \leq\ \pi.\)
\(x =\ −7\ \cos(2(0)) =\ −7(\cos(0)) =\ −7(1) =\ −7\)
\(y =\ −7\ \sin(2(0)) =\ −7(\sin(0)) =\ −7(0) = 0.\) Implies the initial point is \((−7,\ 0).\) For \(t = \pi,\)
\(x =\ −7\ \cos(2\ \pi) =\ −7(1) =\ −7y =\ −7\ \sin(2\ \pi) =\ −7(0) = 0.\) Implies the end point is \((−7,\ 0).\) The initial point and the end point are equal, that is, \((−7,\ 0).\) Hence, the orientation is positive in anticlockwise direction.

0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-02-08
Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
\(\displaystyle{x}=-{7}{\cos{{2}}}{t},{y}=-{7}{\sin{{2}}}{t},{0}\le{t}\le\pi\)
asked 2021-06-06
Find the length of the curve.
\(r(t)=\langle6t,\ t^{2},\ \frac{1}{9}t^{3}\rangle,\ 0\leq t\leq1\)
asked 2021-05-01
Change from rectangular to spherical coordinates. (Let \(\rho\geq 0, 0 \leq \theta \leq 2\pi, \text{ and } 0 \leq \phi \leq \pi\))
(a) (0, -3, 0)
(b) \((-1, 1, -\sqrt{2})\)
asked 2020-12-27

Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: \(x = h + r \cos(?), y = k + r \sin(?)\) Use your result to find a set of parametric equations for the line or conic section. \((When\ 0 \leq ? \leq 2?.)\) Circle: center: (6, 3), radius: 7

asked 2021-05-03
Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:
\(f(x,y)=\begin{cases}xe^{-x(1+y)} & x\geq0\ and\ \geq0\\ 0 & otherwise \end{cases}\)
a) What is the probability that the lifetime X of the first component exceeds 3?
b) What are the marginal pdf's of X and Y? Are the two lifetimes independent? Explain.
c) What is the probability that the lifetime of at least one component exceeds 3?
asked 2021-06-04
Find an equation of the plane tangent to the following surface at the given point.
\(7xy+yz+4xz-48=0;\ (2,2,2)\)
asked 2021-05-12
Evaluate the following integral in cylindrical coordinates triple integral \(\int_{x=-1}^{1}\int_{y=0}^{\sqrt{1-x^{2}}}\int_{z=0}^{2}\left(\frac{1}{1+x^{2}+y^{2}}\right)dzdxdy\)
asked 2021-01-08
Parametric to polar equations Find an equation of the following curve in polar coordinates and describe the curve. \(x = (1 + cos t) cos t, y = (1 + cos t) sin t, 0 \leq t \leq 2\pi\)
asked 2021-05-17
Math the parametric equation with the correct graph
asked 2021-05-14
Use the given graph off over the interval (0, 6) to find the following.

a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)
b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)
c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)
d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)
e) The coordinates of the point of inflection. \((x,\ y)=\)
...