MATH SOLVE

4 months ago

Q:
# Find parametric equations for the path of a particle that moves along the circle x2 + (y − 3)2 = 16 in the manner described. (enter your answer as a comma-separated list of equations. let x and y be in terms of t.) (a) once around clockwise, starting at (4, 3). 0 ≤ t ≤ 2π.

Accepted Solution

A:

Answer:[tex]x(t) = 4 cos(-t)\\y(t) = 4 sin(-t) + 3\\[/tex]and t ∈ [0,2π]Step-by-step explanation:The standard equation for circle is:[tex](x-h) ^2 + (y-k)^2 = r^2[/tex]Comparing our equation with standard[tex]x^2 + (y-3)^2 =16[/tex]h= 0, k= 3r= 4 (as in standard r², so r =√ r²)Required:Parametric equations of the circle Formula:[tex]x = r cos(t) + h\\y= r cos(t) +k[/tex]Putting values of r and h and k we get[tex]x= 4 cos(t) + 0 \\x= 4 cost(t) \\y= 4 sin(t) + 3[/tex]As we need to start the object at (4,3) x(0) = 4 and y(0) = 3and the period is 2πAs the rotation is starting clock wise we will invert the value of t i.e -t[tex]x(t) = 4 cos(-t)\\y(t) = 4 sin(-t) + 3\\[/tex]and t ∈ [0,2π]