Relative lengths of paths and cycles in k-connected graphs

Let G be a k-connected graph where k≥3. It is shown that if G contains a path L of length l then G also contains a cycle of length at least ($\text{(2k ? 4)}\text{(3k ? 4)}$) l. This result is obtained from a constructive proof that G contains 3k² ? 7k + 4 cycles which together cover every edge of L at least 2k² ? 6k + 4 times.