Triangular Plot: Survey Check and Cost Estimate
A land surveyor measures a triangular plot ABC on a sloping site. Side AB = 38.5 m, side AC = 52.0 m, and the included angle BAC = 63°. The owner wants to fence the entire boundary and sow grass seed over the plot. Fencing costs M145 per metre, and grass seed costs M18 per 10 m², sold only in full 10 m² packs; a 5% allowance for waste is added to the seed area before ordering. A neighbour claims that the angle at B is 70° and that therefore the plot is larger than the surveyor's plan. The surveyor must check this claim using the measurements and then prepare a cost estimate. The local council also requires the longest side to be recorded to the nearest 0.1 m and the area to the nearest square metre. The surveyor has a laser measure that reads to 0.1 m, but the angle is measured to the nearest degree.
1. Calculate the length BC to the nearest 0.1 m. Show the formula and substitution.
Read the worked answer
Use the cosine rule: BC² = AB² + AC² − 2(AB)(AC)cos A. BC² = 38.5² + 52.0² − 2(38.5)(52.0)cos63°. 38.5² = 1482.25; 52.0² = 2704; sum = 4186.25. 2 × 38.5 × 52.0 = 4004. cos63° ≈ 0.4540. Product = 4004 × 0.4540 = 1817.82. BC² = 4186.25 − 1817.82 = 2368.43. BC = √2368.43 = 48.666 m, so 48.7 m to nearest 0.1 m.
2. Find the angle ABC and compare it with the neighbour's claim of 70°. Explain why the sine rule alone can give two possible angles here.
Read the worked answer
Use sine rule: sin B / AC = sin A / BC, so sin B = AC sin A / BC = 52.0 × sin63° / 48.666. sin63° = 0.8910; numerator = 46.332; sin B = 46.332 / 48.666 = 0.9520. This gives B = 72.2° or B = 107.8°. The sine rule is ambiguous because sin(180° − B) = sin B. But side AC = 52.0 m is the longest side, so angle B must be the largest angle. If B = 107.8°, then C = 180 − 63 − 107.8 = 9.2°. Using the sine rule for C would give sin C = 38.5 × sin63° / 48.666 = 0.7048, so C = 44.8°, not 9.2°. So B = 72.2°. The neighbour's 70° is close but not correct; the claim underestimates the angle by about 2.2°.
3. Calculate the area of the plot to the nearest square metre, then find the total cost of fencing and grass seed. Remember the 5% seed waste and full 10 m² packs.
Read the worked answer
Area = ½ × AB × AC × sin A = 0.5 × 38.5 × 52.0 × sin63°. 0.5 × 38.5 × 52.0 = 1001.0; × 0.8910 = 891.9 m², so 892 m² to nearest square metre. Add 5% waste: 892 × 1.05 = 936.6 m². Packs needed = 936.6 ÷ 10 = 93.66, so 94 full packs. Seed cost = 94 × 18 = M1,692. Perimeter = 38.5 + 52.0 + 48.7 = 139.2 m. Fencing cost = 139.2 × 145 = M20,184. Total = 20,184 + 1,692 = M21,876.