Answer
a) $x(0)=3ft$, $x(\pi/2)=4ft$ and $x(\pi)=-3ft$.
b) $v(0)=4ft/sec$, $v(\pi/2)=-3ft/sec$ and $v(\pi)=-4ft/sec$.
Work Step by Step
$$x=3\cos t+4\sin t$$
a) Find the particle's position:
- When $t=0$: $$x(0)=3\cos0+4\sin0=3\times1+4\times0=3(ft)$$
- When $t=\pi/2$: $$x(\pi/2)=3\cos\pi/2+4\sin\pi/2=3\times0+4\times1=4(ft)$$
- When $t=\pi$: $$x(\pi)=3\cos\pi+4\sin\pi=3\times(-1)+4\times0=-3(ft)$$
b) To find the particle's velocity, we first need a formula for it, which is the derivative of the position formula:
$$v=x'=(3\cos t+4\sin t)'=-3\sin t+4\cos t$$
- When $t=0$: $$v(0)=-3\sin0+4\cos0=-3\times0+4\times1=4(ft/sec)$$
- When $t=\pi/2$: $$v(\pi/2)=-3\sin\pi/2+4\cos\pi/2=-3\times1+4\times0=-3(ft/sec)$$
- When $t=\pi$: $$v(\pi)=-3\sin\pi+4\cos\pi=-3\times0+4\times(-1)=-4(ft/sec)$$
