Surveying a Triangular Plot
A triangular plot of land has sides of lengths 50 m, 70 m, and 90 m. A surveyor needs to determine the area of the plot and the measure of the largest angle. Additionally, a fence is to be built along the longest side, but a gate of width 3 m must be left open. The surveyor uses the cosine rule and Heron's formula.
1. Calculate the largest angle of the triangle.
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The largest angle is opposite the longest side, which is 90 m. Using the cosine rule: ( \cos C = \frac{a^2 + b^2 - c^2}{2ab} ), where ( c = 90 ), ( a = 50 ), ( b = 70 ). Then ( \cos C = \frac{50^2 + 70^2 - 90^2}{2 \times 50 \times 70} = \frac{2500 + 4900 - 8100}{7000} = \frac{-700}{7000} = -0.1 ). Thus ( C = \arccos(-0.1) \approx 95.74^\circ ). The largest angle is approximately ( 95.7^\circ ).
2. Find the area of the plot using Heron's formula.
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Semi-perimeter ( s = \frac{50+70+90}{2} = 105 ) m. Heron's formula: ( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{105(105-50)(105-70)(105-90)} = \sqrt{105 \times 55 \times 35 \times 15} ). Compute the product: ( 105 \times 55 = 5775 ), ( 35 \times 15 = 525 ), then ( 5775 \times 525 = 3,031,875 ). The square root is ( \sqrt{3,031,875} \approx 1741.23 ) square meters. So the area is approximately 1,741 m².
3. If the fence is to be built along the longest side, and a gate of 3 m is left open, what length of fencing is needed? If fencing costs $25 per meter, what is the total cost?
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The longest side is 90 m. With a 3 m gate, the length of fencing needed is ( 90 - 3 = 87 ) m. The cost is ( 87 \times 25 = 2175 ) dollars. So the fencing will cost $2,175.