Question

Find the length of the curve defined by the vector-valued function r(t) for the given interval: r(t)=⟨t,3cos(t),3sin(t) for −5≤t≤5

A. 15
B. 10
C. 5
D. 20

Answer 1

The length of the curve defined by the vector-valued function r(t) for the given interval -5≤t≤5 is 10, thus the correct option is B.

Step-by-step explanation:

The length of a curve, defined by a vector-valued function, can be calculated by finding the distance of each point along the curve from the origin. This can be done using the formula for finding the magnitude of a vector, which is the square root of the sum of the squared components of the vector.

The distance between the two points (t=-5 and t=5) can be calculated by taking the difference between the two points and then finding the magnitude of the resultant vector. Thus, the difference between the two points is (5 - (-5)) = 10. The magnitude of this vector is 10√3, which is the length of the curve defined by the vector-valued function r(t). Therefore, the length of the curve defined by the vector-valued function r(t) for the given interval -5≤t≤5 is 10.