A pipe closed at end vibrating in the fourth overtone is in unison with the open pipe vibrating in its fourth overtone. The ratio of l_c : l_o is l_c = vibrating length of closed pipe, l_o = vibrating length of open pipe

Step-by-step solution

For a closed pipe, the frequency of the p -th overtone is given by f_c = (2p+1)v 4l_c . For the fourth overtone ( p=4 ), the frequency is f_c = 9v 4l_c . For an open pipe, the frequency of the p -th overtone is given by f_o = (p+1)v 2l_o . For the fourth overtone ( p=4 ), the frequency is f_o = 5v 2l_o . Since the two pipes are in unison, their frequencies are equal: 9v 4l_c = 5v 2l_o l_c l_o = 9 2 4 5 = 18 20 = 9 10 The ratio l_c : l_o is 9 : 10 . Answer: 9 : 10