Answer
(a) $0.8$
(b) $y(t)=3e^{-0.8t}cos(330\pi t)$.
Work Step by Step
Identify the given quantities as $y(0)=|a(0)|=3cm, f=165Hz, |a(2)|=0.6cm$
(a) The general form of the amplitude is given by $|a(t)|=|ke^{-ct}|$, use the conditions given above, we have |ke^0|=3 which gives $k=3$ (assuming it started with a positive displacement). The condition $ |a(2)|=0.6cm$ gives $3e^{-2c}=0.6$ and we have $-2c=ln0.2$ which gives $c=ln5/2\approx0.8$
(b) Model the equation with a damping cosine model $y(t)=ke^{-ct}cos(\omega t)$ which satisfies the initial condition $y(0)=3cm$ . With the above results, we have $\omega=2\pi f=330\pi$ and $y(t)=3e^{-0.8t}cos(330\pi t)$.
