Correct Answer - Option 3 : 10 minutes

\(Heat,\;H = {I^2}Rt = \frac{{{V^2}}}{R}t\)

For same voltage and heat: \(\frac{t}{R} = constant\;and\;R = \frac{{\rho l}}{A}\;so\frac{R}{l} = constant\)

\(\frac{{{t_1}}}{{{R_1}}} = \frac{{{t_2}}}{{{R_2}}} \Rightarrow \frac{{{t_1}}}{{{t_2}}} = \frac{{{R_1}}}{{{R_2}}} = \frac{{{l_1}}}{{{l_2}}} \Rightarrow {t_2} = \frac{{{l_2}}}{{{l_1}}}\; \times \;{t_1} = \frac{2}{3}\; \times \;15 = 10\;minutes\;\)