An organ pipe p_1 closed at one end has vibrating air column in its...

For pipe $P_1$ (closed at one end), the frequency of the first overtone corresponds to the third harmonic. Its frequency is given by:

$f_1 = \dfrac{3v}{4L_1}$

For pipe $P_2$ (open at both ends), the frequency of the third overtone corresponds to the fourth harmonic. Its frequency is given by:

$f_2 = \dfrac{4v}{2L_2}$

Since both pipes are in resonance with the same tuning fork, their frequencies must be equal:

$f_1 = f_2$

$\dfrac{3v}{4L_1} = \dfrac{4v}{2L_2}$

$\dfrac{3}{4L_1} = \dfrac{2}{L_2}$

$\dfrac{L_1}{L_2} = \dfrac{3}{8}$

The ratio of the length of pipe $P_1$ to that of $P_2$ is $3 : 8$.

Answer: $3 : 8$