To determine if the function \( x(t) = \sin(3t) + \sin(5t) \) is periodic and find its fundamental period if it is, we follow these steps:

1. Recall that a signal is periodic if it repeats itself after a certain period \( T \). Mathematically, a continuous-time signal \( x(t) \) is periodic if there exists a positive value \( T \) such that:

   \( x(t) = x(t + T) \)

2. The sine functions \( \sin(3t) \) and \( \sin(5t) \) are individually periodic functions with periods:

   • The period of \( \sin(3t) \) is \( T_1 = \frac{2\pi}{3} \).
   • The period of \( \sin(5t) \) is \( T_2 = \frac{2\pi}{5} \).

3. To find out if their sum is periodic, we check the ratio of their fundamental frequencies:

   Let \( \omega_1 = 3 \) and \( \omega_2 = 5 \). The ratio \(\frac{\omega_1}{\omega_2}\) is \(\frac{3}{5} \), which is a rational number.

4. Since the ratio of the frequencies is rational, the signal \( x(t) \) is periodic. The fundamental period \( T \) of the sum of two signals is the least common multiple (LCM) of their individual periods:

   \( T = \text{LCM}\left( \frac{2\pi}{3}, \frac{2\pi}{5} \right) \)

5. The LCM of \(\frac{2\pi}{3}\) and \(\frac{2\pi}{5}\) is given by:

   \( T = \frac{2\pi \times \text{LCM}(3, 5)}{15} = \frac{2\pi}{1} = 2\pi \)

Therefore, the signal \( x(t) = \sin(3t) + \sin(5t) \) is periodic with a fundamental period of \( T = 2\pi \). Thus, the correct answer is T=2\pi.