(a) To find the probability that an apple is sold to the supermarket, we need to calculate the probability that the weight of an apple is between 142 grams and 205 grams. We use the standard normal distribution:

\(P(142 < X < 205) = P\left(\frac{142 - 170}{25} < z < \frac{205 - 170}{25}\right)\)

This simplifies to:

\(P(-1.12 < z < 1.4)\)

Using the standard normal distribution table, we find:

\(\Phi(1.4) - (1 - \Phi(1.12)) = 0.9192 + 0.8686 - 1\)

Thus, the probability is:

\(0.788\)

(b) To calculate the total income, we first find the probability that an apple weighs more than 205 grams:

\(P(X > 205) = 1 - 0.9192 = 0.0808\)

The income from apples sold to the supermarket is:

\(0.788 \times 0.24 \times 20000\)

The income from apples sold to the local shop is:

\(0.0808 \times 0.30 \times 20000\)

Therefore, the total income is:

\((0.0808 \times 0.30 + 0.788 \times 0.24) \times 20000 = 4266.24\)