Two objects are traveling in elliptical paths given by the following parametric
ID: 2882035 • Letter: T
Question
Two objects are traveling in elliptical paths given by the following parametric equations: First Object: x_1 = 4 cos(t) y_1 = 2sin(t) Second Object: x_2 = 2 sin(2t) y_2 = 3 cos(2t) where t is measured in seconds and x_1, y_1, x_2, and y_2 are measured in meters. The distance, s, between the two objects is given by the function s(x_1, y_1, x_2, y_2) = Squareroot (x_2 - x_1)^2 + (y_2 - y_1)^2 Use the chain rule for functions of several variables to find ds/dt, the rate that the distance between the two objects is changing, when t = pi. Present your work in an organized fashion and be sure to use correct units on your final answer.Explanation / Answer
given x1=4cos(t) ,x2=2sin(2t) ,y1=2sin(t),y2=3cos(2t)
s=[(x2-x1)2+(y2-y1)2]
s/x1 =(1/2[(x2-x1)2+(y2-y1)2])*(2(x2-x1)(0-1)+0)=(-(x2-x1)/[(x2-x1)2+(y2-y1)2])
s/x2 =(1/2[(x2-x1)2+(y2-y1)2])*(2(x2-x1)(1-0)+0)=((x2-x1)/[(x2-x1)2+(y2-y1)2])
s/y1 =(1/2[(x2-x1)2+(y2-y1)2])*(0+2(y2-y1)(0-1))=(-(y2-y1)/[(x2-x1)2+(y2-y1)2])
s/y2 =(1/2[(x2-x1)2+(y2-y1)2])*(0+2(y2-y1)(1-0))=((y2-y1)/[(x2-x1)2+(y2-y1)2])
dx1/dt =-4sin(t) ,dx2/dt =4cos(2t) ,dy1/dt=2cos(t),dy2/dt=-6sin(2t)
at t=
x1=-4 ,x2=0 ,y1=0,y2=3
s/x1 =(-(0-(-4))/[(0-(-4))2+(3-0)2])= -4/5
s/x2 =4/5
s/y1 =-3/5
s/y2 =3/5
dx1/dt =0 ,dx2/dt =4,dy1/dt=2,dy2/dt=0
ds/dt =(s/x1*dx1/dt) +(s/x2*dx2/dt) +(s/y1*dy1/dt) +(s/y2*dy2/dt)
ds/dt =(-4/5*0) +(4/5*4) +(-3/5*2) +(3/5*0)
ds/dt =0 +(16/5) +(-6/5)+0
ds/dt =10/5
ds/dt =2 meters per second