Find a path that traces the circle in the plane y=5 with radius r=2 and center (−1, 5, −5) with conctant speed 8. r1(s)=⟨ ―⟩

Question

Find a path that traces the circle in the plane y=5 with radius r=2 and center (−1, 5, −5) with conctant speed 8.  r1(s)=⟨ ―⟩

Bromodermgs · Accepted Answer

Step 1The radius if circle&amp;nbsp;=2&amp;nbsp;and center&amp;nbsp;(−1,&amp;nbsp;5,&amp;nbsp;−4)Such a circle&#039;s equation is given by:(x−(−1))2+(z−(−4))2=22,&amp;nbsp;y=5(x+1)2+(z+4)2=4,&amp;nbsp;y=5The circle&#039;s parametric equations are provided by:x+1=2cos⁡ωt,&amp;nbsp;y=5,&amp;nbsp;z+4=2sin⁡ωx=−1+2cos⁡ωt,&amp;nbsp;y=5,&amp;nbsp;z=−4+2sin⁡ωSpeed is given by:v=(&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;)2+(&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dt&amp;nbsp;)2+(&amp;nbsp;dz&amp;nbsp;&amp;nbsp;dt&amp;nbsp;)2Compute the derivatives:&amp;nbsp;dx&amp;nbsp;&amp;nbsp;dt&amp;nbsp;=0+2ω(−sin⁡ωt)=−2ωsin⁡ω&amp;nbsp;dy&amp;nbsp;&amp;nbsp;dt&amp;nbsp;=0&amp;nbsp;dz&amp;nbsp;&amp;nbsp;dt&amp;nbsp;=0+2ω(cos⁡ωt)=2ωcos⁡ωIn given question we have, speed&amp;nbsp;=8Substitute these derivatives and speed value in the speed equation above:8=(−2ωsin⁡ωt)2+(0)2+(2ωcos⁡ωt)28=4ω2sin2ωt+4ω2cos2ωt8=4ω2(sin2ωt+cos2ωt}9=4ω2(1)8=2ω82=ω4=ωTherefore, the parametric equations are :x=−1+2cos⁡4t,&amp;nbsp;y=5,&amp;nbsp;z=−4+2sin⁡4tThe path is given by:r(t)=⟨−1+&amp;nbsp;

Find a path that traces the circle in the plane

Answered question

Answer & Explanation

New Questions in Calculus 2