Quadrilateral Kitchen Garden Quotation
The headmaster of a government high school in Cuttack has marked a four-sided plot ABCD for a kitchen garden. The surveyor's field book records AB = 13 m, BC = 14 m, CD = 15 m, DA = 20 m, and diagonal AC = 21 m. A local contractor quotes ₹85 per metre for barbed-wire fencing along the boundary and ₹40 per square metre for levelling the plot. The headmaster has ₹18,000 from the school development fund and wants to keep 8% of the estimated total as contingency. Before approving the work, the school committee asks whether the plot can be treated as two triangles for area calculation, whether the contractor's quotation fits the budget, and what happens if the contractor offers a 5% discount on fencing only. The committee is also worried that the diagonal chosen might overstate the area.
1. Show that diagonal AC divides ABCD into two valid triangles, and calculate the area of the plot using Heron's formula. Round the area to two decimal places.
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Check the two triangles:
- Triangle ABC: 13 + 14 = 27 > 21, and the other two sums are also larger, so it is valid.
- Triangle ACD: 15 + 20 = 35 > 21, so it is valid.
For ABC, s = (13+14+21)/2 = 24 m. Area = sqrt(24(24-13)(24-14)(24-21)) = sqrt(24×11×10×3) = sqrt(7920) ≈ 88.99 m².
For ACD, s = (21+15+20)/2 = 28 m. Area = sqrt(28(28-21)(28-15)(28-20)) = sqrt(28×7×13×8) = sqrt(20384) ≈ 142.77 m².
Total area ≈ 88.99 + 142.77 = 231.76 m².
2. Using the area from Task 1, compute fencing cost, levelling cost, the 8% contingency, and decide whether ₹18,000 is sufficient.
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Perimeter = 13+14+15+20 = 62 m. Fencing = 62 × 85 = ₹5,270. Levelling = 231.76 × 40 = ₹9,270.40. Subtotal = 5,270 + 9,270.40 = ₹14,540.40. Contingency = 8% of 14,540.40 = ₹1,163.23. Grand total ≈ ₹15,703.63. Since ₹15,703.63 < ₹18,000, the quotation fits the budget, leaving about ₹2,296.37.
3. The contractor offers 5% off fencing only. Recalculate the total with contingency and state the saving compared with the original quotation.
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Discount is 5% of fencing only: 0.05 × 5,270 = ₹263.50. New subtotal = 14,540.40 − 263.50 = ₹14,276.90. New contingency = 8% of 14,276.90 = ₹1,142.15. New grand total ≈ ₹15,419.05. Saving = 15,703.63 − 15,419.05 = ₹284.58. So the discount saves only about ₹285 after contingency, not ₹263.50.