(a) Initial Heights:

(i) Plant B:
\( h_B(0) = 8(0) + 32 = \boxed{32 \text{ cm}} \)

(ii) Plant A:
\( h_A(0) = \sin(6) + 27 \)
\( \sin(6 \text{ rad}) \approx -0.2794 \)
\( \approx -0.2794 + 27 = \boxed{26.7 \text{ cm}} \) (3 s.f.)

(b) Height Intersection Points:

Solve \( \sin(2t + 6) + t – 5 = 0 \)
Numerical solutions in \( [4,6] \):
\( t \approx \boxed{4.01} \), \( \boxed{4.70} \), and \( \boxed{5.88} \) weeks
Method: Graphical analysis and intermediate value theorem

(c) Height Comparison Proof:

Consider \( D(t) = h_A(t) – h_B(t) = \sin(2t + 6) + t – 5 \)
Since \( \sin(2t + 6) \geq -1 \):
\( D(t) \geq -1 + t – 5 = t – 6 \)
For \( t > 6 \): \( t – 6 > 0 \Rightarrow D(t) > 0 \)
Thus \( h_A(t) > h_B(t) \) for all \( t > 6 \) weeks.

(d) Growth Rate Analysis:

Derivatives:
\( h_A'(t) = 2\cos(2t + 6) + 9 \)
\( h_B'(t) = 8 \)

Solve \( h_B'(t) > h_A'(t) \):
\( 8 > 2\cos(2t + 6) + 9 \)
\( \cos(2t + 6) < -0.5 \)

Solution Intervals:
1. \( 1.189 < t < 2.236 \) weeks
2. \( 4.330 < t < 5.378 \) weeks
3. \( 7.472 < t < 8.519 \) weeks

Total Duration:
Each interval spans \( \frac{\pi}{3} \) weeks
\( 3 \times \frac{\pi}{3} = \boxed{\pi \text{ weeks}} \) (\( \approx 3.14 \) weeks)