A gong makes a loud noise when struck. The noise gradually gets less and less loud until it fades below the sensitivity of the human ear. The simplest model of how the gong produces the sound we hear treats the gong as a damped harmonic oscillator. The tone we hear is related to the frequency \(f\) of the oscillation, and its loudness is proportional to the energy of the oscillation.

(a) If the loudness drops to \(85 \%\) of its original value in \(4.0 \mathrm{~s}\), what is the time constant of the damped oscillation?

(b) How long does it take for the sound to be \(25 \%\) as loud as it was at the start?

(c) What fraction of the original loudness remains after \(1.0 \mathrm{~min}\) ?