The probability of an apple being underweight is given by \(p = \frac{2}{15}\).

For a sample of 200 apples, the mean \(\mu\) and variance \(\sigma^2\) are calculated as follows:

\(\mu = 200 \times \frac{2}{15} = 26.67\)

\(\sigma^2 = 200 \times \frac{2}{15} \times \frac{13}{15} = 23.11\)

We need to find \(P(21 < X < 35)\).

Using continuity correction, this becomes \(P(21.5 < X < 34.5)\).

Standardizing, we have:

\(P\left( \frac{21.5 - 26.67}{\sqrt{23.11}} < z < \frac{34.5 - 26.67}{\sqrt{23.11}} \right)\)

\(P(-1.075 < z < 1.629)\)

Using the standard normal distribution table:

\(P(z < 1.629) = 0.9483\)

\(P(z < -1.075) = 0.1411\)

Thus, \(P(-1.075 < z < 1.629) = 0.9483 - 0.1411 = 0.807\).