Question

Determine whether the following curve uses arc length as a parameter. If not, find a description that uses arc length as a parameter. r(t) = (4t^2,8t^2, root 89t^2), for 1 leq t leq 4 r(t) does not use arc length as a parameter. A description of the curve that uses arc length as a parameter is r(s) = (Type exact answers, using radicals as needed.)

Answer 1

Answer:

solution:

If ║v(t)║= 1, for all t >= a, then the parameter t corresponds to arc length.  

║v║=║r'║*║v║=√4t^2+16t^2+64t^2=√84t=2√21t is not equal to 1,

so it doesn't use arc length as a parameter.  

Consider s =  √21t

    r(s) =(s/√21, 2s/√21, 4s/√21) for √21 <=s<=16√21

║r'(s)║=(1/√21)*√1+4+16

          = 1

so now it does use arc length as its parameter.