The Ambiguous Survey of a Triangular Plot
A surveyor is measuring a triangular plot ABC for a fence. She measures AB = 85 m, AC = 62 m, and angle ABC = 38°. She uses the sine rule to find angle ACB and obtains 57.6°, then calculates BC = 100.2 m and orders 101 m of fencing for that side. A second surveyor says the measurement is under-specified and that another triangle with the same three given measurements is possible. The plot must be fenced along all three sides, and the supplier has only 40 m of the required BC fencing in stock. The owner also insists that the angle at C must be obtuse to match the neighbouring property boundary. The first surveyor's calculator is in degree mode, and she has not checked whether the inverse sine could have a second solution. The fence supplier charges $18.50 per metre and rounds orders up to the next whole metre.
1. Use the sine rule to show that angle ACB has two possible values between 0° and 180°. Calculate both values to one decimal place.
Read the worked answer
In triangle ABC, the side opposite angle C is AB = 85 m, and the side opposite angle B is AC = 62 m. The sine rule gives
sin C / 85 = sin 38° / 62,
so sin C = 85 sin 38° / 62 ≈ 85×0.615661/62 ≈ 0.84405.
The inverse sine gives C ≈ 57.6°. Because sin(180°−C) = sin C, the second possible value is C ≈ 180°−57.6° = 122.4°. Both are between 0° and 180°, and both leave a positive angle at A: 84.4° in the first case and 19.6° in the second. Therefore the given data are ambiguous.
2. For each possible value of angle ACB, calculate the corresponding length BC. Then decide which triangle satisfies the owner's condition that angle C is obtuse and the supplier's stock limit of 40 m for BC. Justify your choice.
Read the worked answer
For C = 57.6°, A = 180°−38°−57.6° = 84.4°. By the sine rule,
BC / sin 84.4° = 62 / sin 38°,
so BC = 62 sin 84.4° / sin 38° ≈ 100.2 m.
For C = 122.4°, A = 180°−38°−122.4° = 19.6°. Then
BC = 62 sin 19.6° / sin 38° ≈ 33.7 m.
The owner requires angle C obtuse, so C ≈ 122.4°. This gives BC ≈ 33.7 m, which is under the 40 m stock limit. The first surveyor's acute solution C ≈ 57.6° gives BC ≈ 100.2 m, which is not obtuse and exceeds the stock limit. The correct interpretation is therefore C ≈ 122.4° and BC ≈ 33.7 m.
3. The supplier sells fencing in whole metres, rounded up, at $18.50 per metre. What is the cost of the correct BC section, and how much did the first surveyor over-order in metres and dollars by using the acute solution?
Read the worked answer
Correct BC ≈ 33.7 m, rounded up to 34 m. Cost = 34×18.50 = $629.00.
First surveyor's BC ≈ 100.2 m, rounded up to 101 m. Cost = 101×18.50 = $1,868.50.
Over-order = 101−34 = 67 m. Extra cost = 1868.50−629.00 = $1,239.50. So the first surveyor over-ordered by 67 m, costing $1,239.50 more.