Explanation

(c) $H = \frac{V^{2}}{R}t$
Since supply voltage is same and equal amount of heat is produced, therefore
$\frac{R_{1}}{t_{1}} = \frac{R_{2}}{t_{2}}\ or\ \frac{R_{1}}{R_{2}} = \frac{t_{1}}{t_{2}}\ \ \ \ \ldots(i)$
But$\ R \propto l \Rightarrow \frac{R_{1}}{R_{2}} = \frac{l_{1}}{l_{2}}\ \ \ $…(ii)
By (i) and (ii)$,\ \frac{l_{1}}{l_{2}} = \frac{t_{1}}{t_{2}}\ \ $…(iii)
Now$\ l_{2} = \frac{2}{3}l_{1} \Rightarrow \frac{l_{1}}{l_{2}} = \frac{3}{2}$
$\therefore$ By equation$\ (iii),\frac{3}{2} = \frac{15}{t_{2}} \Rightarrow t_{2} = 10\ minutes$