(a) For a project consisting of several activities, the allotted time and the dependencies of the activities are presented below…

(a) CPM network, critical path and float

(i) Network (activity-on-node): P and Q have no predecessor. P is followed by S and T; Q is followed by R; U follows T and R. So the paths are P–S, P–T–U and Q–R–U.

Forward pass (ES, EF): P 0–5, Q 0–4, R 4–10, S 5–10, T 5–12, U max(12, 10) = 12 to 16. Project duration = 16 days.

Path lengths: P–T–U = 5 + 7 + 4 = 16; Q–R–U = 4 + 6 + 4 = 14; P–S = 10. Critical path: P → T → U, 16 days.

(ii) Backward pass (LF, LS), with S and U ending at 16: U 12–16; T 5–12; R 6–12; S 11–16; P: LF = min(11, 5) = 5, LS = 0; Q: LF = 6, LS = 2.

Total float = LS − ES:

P: ES 0, EF 5, LS 0, LF 5, float 0. Q: ES 0, EF 4, LS 2, LF 6, float 2. R: ES 4, EF 10, LS 6, LF 12, float 2. S: ES 5, EF 10, LS 11, LF 16, float 6. T: ES 5, EF 12, LS 5, LF 12, float 0. U: ES 12, EF 16, LS 12, LF 16, float 0.

The zero-float activities P, T and U confirm the critical path.

(b) Equilibrium cant and maximum permissible speed

For a 2° curve, R = 1750/D = 1750/2 = 875 m. For broad gauge the dynamic gauge is G = 1750 mm, so

Equilibrium cant e = GV²/(127R) = 1750 × 80²/(127 × 875) = 100.8 mm (about 10 cm).

This is the cant provided on the track. The maximum cant deficiency for broad gauge is 75 mm. At the maximum speed the equilibrium cant needed is e + C_d = 100.8 + 75 = 175.8 mm.

V_max² = 175.8 × 127 × 875/1750 = 11 163, so V_max ≈ 105.7 km/h, about 105 km/h. (The IR form V = 0.27√[(C_a + C_d)R] = 105.9 km/h agrees.)

(c)(i) Photograph scale and exposure interval

Flying height above ground H − h = 5000 − 2000 = 3000 m. Scale = f/(H − h) = 0.300/3000 = 1 : 10 000.

Ground coverage along the flight line = 0.300 m × 10 000 = 3000 m. With 65% overlap, the air base = (1 − 0.65) × 3000 = 1050 m between consecutive exposure stations.

(c)(ii) Spectral reflectance curve

A spectral reflectance curve plots the percentage of incident radiation that a surface reflects against wavelength. It is the spectral signature of the material.

Its significance in remote sensing: different features reflect differently, so the curve allows them to be told apart. Healthy vegetation has low reflectance in the blue and red bands, a small peak in the green, and a sharp rise in the near infrared. Clear water reflects little and almost nothing beyond the near infrared. Dry soil shows a steady rise with wavelength. The curves guide the choice of sensor bands, support classification and indices such as NDVI, and allow monitoring of crop stress, water quality and land-cover change.