The weights of the apples are normally distributed with mean \(\mu = 182\) and standard deviation \(\sigma = 20\). We need to find \(w\) such that \(P(X > w) = 0.72\).

First, convert the problem to a standard normal distribution problem:

\(P\left(Z > \frac{w - 182}{20}\right) = 0.72\)

This implies:

\(P\left(Z < \frac{w - 182}{20}\right) = 0.28\)

Using the standard normal distribution table, find the z-value for which \(P(Z < z) = 0.28\). This gives \(z = -0.583\).

Now, equate and solve for \(w\):

\(\frac{w - 182}{20} = -0.583\)

\(w - 182 = -0.583 \times 20\)

\(w = 182 - 11.66\)

\(w = 170.34\)

Therefore, \(170 \leq w < 170.35\).