(i) To find the proportion of large pineapples, calculate \(P(X > 570)\):

Standardize: \(z = \frac{570 - 500}{91.5} = 0.7650\)

\(P(Z > 0.7650) = 1 - P(Z < 0.7650) = 1 - 0.7779 = 0.222\)

To find the proportion of small pineapples, calculate \(P(X < 390)\):

Standardize: \(z = \frac{390 - 500}{91.5} = -1.202\)

\(P(Z < -1.202) = 0.1147\)

The proportion of medium pineapples is \(1 - (0.222 + 0.115) = 0.663\).

(ii) To find the weight exceeded by the heaviest 5% of pineapples, find \(x\) such that \(P(X > x) = 0.05\).

Find the critical value: \(z = 1.645\)

Standardize: \(1.645 = \frac{x - 500}{91.5}\)

\(x = 1.645 \times 91.5 + 500 = 651\)

(iii) To find \(k\) such that \(P(k < X < 610) = 0.3\):

\(P(X > 610) = 0.1147\) (from symmetry)

\(0.3 + 0.1147 = 0.4147 \Rightarrow \Phi(x) = 0.5853\)

Find \(z\) such that \(\Phi(z) = 0.5853\): \(z = 0.215\)

Standardize: \(0.215 = \frac{k - 500}{91.5}\)

\(k = 0.215 \times 91.5 + 500 = 520\)