CPM Homework Help : APCALC Problem 12-110

Multiple Choice: A particle moves in the plane according to the set of parametric equations $x(t) = \cos(π t)$ and $y ( t ) = \frac { t ^ { 2 } } { t + 1 }$ . What is the magnitude of the velocity vector at $t =\frac { 1 } { 2 }$ ?

$\frac { 1 } { 6 }$

$π +\frac { 5 } { 9 }$

$\frac { \sqrt { 106 } } { 9 }$

$\frac { \sqrt { 81 \pi ^ { 2 } + 25 } } { 9 }$

Step 1:

$\vec{v}(t)=\langle -\pi\sin(\pi t),\frac{t^2+2t}{(t+1)^2}\rangle$

Step 2:

$\vec{v}(0.5)=\langle -\pi\sin(\pi/2),\frac{0.5^2+1}{(1.5)^2}\rangle=\langle -\pi,\frac{5}{9}\rangle$

Step 3:

Sketch this vector, then use the Pythagorean Theorem to determine its magnitude.