First, we standardize the lengths using the formula for the standard normal variable:

\(z = \frac{x - \mu}{\sigma}\)

For rods shorter than 15.75 cm:

\(P(x < 15.75) = P\left(z < \frac{15.75 - 16}{0.2}\right) = P(z < -1.25)\)

Using symmetry and standard normal tables, \(P(z < -1.25) = 1 - P(z < 1.25) = 1 - 0.8944 = 0.1056\).

For rods longer than 16.25 cm:

\(P(x > 16.25) = P\left(z > \frac{16.25 - 16}{0.2}\right) = P(z > 1.25) = 0.1056\) (by symmetry).

The probability of a rod being usable is:

\(P(\text{usable}) = 1 - P(x < 15.75) - P(x > 16.25) = 1 - 0.1056 - 0.1056 = 0.7888\).

The expected number of usable rods in a batch of 1000 is:

\(1000 \times 0.7888 = 788.8\).

Rounding gives approximately 788 or 789 usable rods.