Question

I’m having trouble with this problem. Please help

Answer 1

The software rendering this question really butchered it... I think it's saying the displacement vector for the particle is

$\vec r(t) = \sin(t) \, \vec\imath + 4t^3 \, \vec\jmath + 5t \, \vec k$

Differentiate twice with respect to time $t$ to get the velocity and acceleration vectors, respectively:

$\dfrac{d\vec r}{dt} = \vec v(t) = \cos(t) \,\vec\imath + 12t^2 \,\vec\jmath + 5 \,\vec k$

$\dfrac{d^2\vec r}{dt^2} = \dfrac{d\vec v}{dt} = \vec a(t) = -\sin(t) \,\vec\imath + 24t \,\vec\jmath$

Then at $t=3$, the acceleration of the particle is

$\vec a(3) = -\sin(3)\,\vec\imath + 72\,\vec\jmath$

which most closely resembles the first choice.