1. Identify the beat form

The resultant wave is given as $x=a\cos(1.5t)\cos(50.5t).$

For two harmonic waves of nearly equal angular frequencies $\omega_1$ and $\omega_2$ moving in the same direction, $x=2A\cos\left(\frac{\omega_1-\omega_2}{2}t\right)\cos\left(\frac{\omega_1+\omega_2}{2}t\right).$

Comparing with $x=a\cos(1.5t)\cos(50.5t),$ we get $\frac{\omega_1-\omega_2}{2}=1.5.$

So, $\omega_1-\omega_2=3\ \text{rad/s}.$

2. Find the beat angular frequency

The amplitude varies as $\cos(1.5t).$

Beats are heard/intensity maxima repeat with beat angular frequency $\omega_b=\omega_1-\omega_2=3\ \text{rad/s}.$

Hence the beat frequency is $f_b=\frac{\omega_b}{2\pi}=\frac{3}{2\pi}\ \text{Hz}.$

3. Find beat period

Beat period is $T_b=\frac{1}{f_b}=\frac{2\pi}{3}\ \text{s}.$

Numerically, $T_b\approx \frac{6.283}{3}\approx 2.09\ \text{s}.$

Nearest integer: $T_b\approx 2\ \text{s}.$

4. Choose the correct option

So the correct option is:

$\boxed{\text{C: }2\text{ s}}$