Answer
0.46
Work Step by Step
Step 1. Write the general form of the damped harmonic motion model as $y(t)=ke^{-ct}cos(\omega t)$, and
the damped amplitude is given by $|a(t)|=|ke^{-ct}|$
Step 2. Based on the conditions given, we have $|a(3)|=\frac{1}{4}|a(0)|$ which leads to $|ke^{-3c}|=\frac{1}{4}|ke^{0}|$
Step 3. Solve the above equation as $3c=ln4$ which gives $c=ln4/3\approx0.46$
