jmc

Of the teenage ages, 13, 17, and 19 are prime; $14=2\cdot 7$, $15=3\cdot 5$, $16=2^4$, and $18=2\cdot 3^2$. The prime factorization of 705,600 is $2^6\cdot 3^2\cdot 5^2\cdot 7^2$. Since $7^2|705600$, 2 people must be 14 years old (which takes care of $2^2\cdot 7^2$). Similarly, since $5^2|705600$, 2 people must be 15 years old (which takes care of $3^2\cdot 5^2$). $2^4$ remains, which means that 1 person is 16. Therefore, the mean of the ages is $\frac{2\cdot 14+2\cdot 15+16}{5}=\frac{74}{5}=\boxed{14.8}$ years.