(a) The network of a certain project is shown in the figure given below with the estimated time duration (in days) of various…

(a) Activities (days): 1–2 = 6, 1–3 = 8, 2–5 = 7, 3–4 = 7, 3–6 = 10, 4–5 = 6, 4–7 = 9, 5–7 = 8, 6–7 = 9, 7–8 = 6.

(i) Earliest event times (forward pass, maximum at merge events):

E₁ = 0; E₂ = 6; E₃ = 8; E₄ = 8 + 7 = 15; E₅ = max(6 + 7, 15 + 6) = 21; E₆ = 8 + 10 = 18; E₇ = max(21 + 8, 15 + 9, 18 + 9) = 29; E₈ = 29 + 6 = 35 days.

(ii) Latest event times (backward pass from L₈ = 35, minimum at burst events):

L₈ = 35; L₇ = 29; L₆ = 29 − 9 = 20; L₅ = 29 − 8 = 21; L₄ = min(21 − 6, 29 − 9) = 15; L₃ = min(15 − 7, 20 − 10) = 8; L₂ = 21 − 7 = 14; L₁ = min(14 − 6, 8 − 8) = 0.

(iii)–(v) Activity times (EST = E_i, EFT = EST + t, LFT = L_j, LST = LFT − t, total float TF = LST − EST):

• 1–2: EST 0, EFT 6, LST 8, LFT 14, TF 8
• 1–3: EST 0, EFT 8, LST 0, LFT 8, TF 0
• 2–5: EST 6, EFT 13, LST 14, LFT 21, TF 8
• 3–4: EST 8, EFT 15, LST 8, LFT 15, TF 0
• 3–6: EST 8, EFT 18, LST 10, LFT 20, TF 2
• 4–5: EST 15, EFT 21, LST 15, LFT 21, TF 0
• 4–7: EST 15, EFT 24, LST 20, LFT 29, TF 5
• 5–7: EST 21, EFT 29, LST 21, LFT 29, TF 0
• 6–7: EST 18, EFT 27, LST 20, LFT 29, TF 2
• 7–8: EST 29, EFT 35, LST 29, LFT 35, TF 0

(vi) The zero-float activities form the critical path 1–3–4–5–7–8, duration 8 + 7 + 6 + 8 + 6 = 35 days.

(b) Rate analysis for 10 m² (quantities per 10 m², then reduced to 1 m²). Item: 25 mm thick cement concrete 1:2:4 (cement : coarse sand : 10–12.5 mm graded stone aggregate) floor, finished with a floating coat of neat cement. Market rates are assumed.

Materials:

• Wet volume = 10 × 0.025 = 0.25 m³; dry volume = 1.52 × 0.25 = 0.38 m³ (sum of ratio = 7).
• Cement = 0.38/7 = 0.0543 m³ = 78 kg ≈ 1.56 bags; neat cement floating coat ≈ 2.2 kg/m² = 22 kg = 0.44 bag; total ≈ 2.0 bags × ₹400 = ₹800.
• Coarse sand = 2 × 0.38/7 = 0.109 m³ × ₹1,800 = ₹196.
• Stone aggregate 10–12.5 mm = 4 × 0.38/7 = 0.217 m³ × ₹1,600 = ₹347.
• Materials total = ₹1,343.

Labour (for 10 m²):

• Mason 1.0 day × ₹900 = ₹900.
• Mazdoor (beldar) 1.5 days × ₹650 = ₹975.
• Bhisti 0.5 day × ₹700 = ₹350.
• Labour total = ₹2,225.

Rate:

• Materials + labour = ₹3,568.
• Water charges @ 1% = ₹36; tools, plant and sundries @ 1% = ₹36; sub-total = ₹3,640.
• Contractor's profit and overheads @ 15% = ₹546.
• Cost of 10 m² = ₹4,186, so rate ≈ ₹419 per m² (excluding GST).

Notes: the aggregate size is kept to about one-third of the 25 mm thickness; the floor is laid in panels, tamped, floated with neat cement while green and cured for at least 7 days. The rate changes in proportion to the local material and labour rates.

(c) A diamond crossing is provided where two tracks cross at an angle. Its four rail intersections form a rhombus: two acute crossings at the ends of the long diagonal (each with a nose, wing rails and check rails) and two obtuse crossings at the ends of the short diagonal (each with point rails, knuckle rails and check rails). Sleepers are common to both tracks and laid square to the long diagonal.

Design for two B.G. tracks (G = 1.676 m) crossing at 1 in 8.5:

• Crossing angle α = cot⁻¹ 8.5 = 6°42′35″; obtuse crossing angle = 180° − α = 173°17′25″; α/2 = 3°21′17″.
• Side of the diamond (length along each track between its crossings) = G cosec α = 1.676 × 8.559 = 14.34 m.
• Long diagonal (between the acute crossings) = G cosec(α/2) = 1.676/0.0585 = 28.64 m.
• Short diagonal (between the obtuse crossings) = G sec(α/2) = 1.676 × 1.0017 = 1.68 m.
• 1 in 8.5 is the flattest diamond crossing permitted on B.G.: as the angle flattens, the unguarded gap opposite the obtuse crossings becomes longer and a wheel may take the wrong flangeway, so check rails are raised and clearances kept tight at the obtuse crossings.