Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number.
We begin by listing the two-digit primes with 1 as the tens digit:

11, 13, 17, 19.

When reversed, the above numbers are 11, 31, 71, and 91.  The first three are prime, but 91 is composite (7 times 13), as desired.  Hence, our desired prime is $\boxed{19}$.