A particle moves in the yz-plane along the curve represented by the vector-valued function . Find the minimum value of .
A) 6
B) 4
C) 7
D) 9
E) 8

Question

A particle moves in the yz-plane along the curve represented by the vector-valued function  . Find the minimum value of  .
A) 6 
B) 4 
C) 7 
D) 9 
E) 8

Quizplus · Accepted Answer

The magnitude of a vector-valued function r(t) represents the distance the particle travels along its path, so we must minimize ||r(t)||. 
||r(t)|| = √(10^2 + (4sin(t))^2 + (4cos(t)-3)^2) = √(116 - 24cos(t)) 
To minimize this expression, we need to maximize cos(t), which means t=0. Hence, the minimum value of ||r(t)|| is √92 = 2√23. Finally, we have ||r(t)||^2 = 92, so the correct answer is B since it is the only integer value close enough to the square root of 92.

A Particle Moves in the Yz-Plane Along the Curve Represented