Problem 1 :
A researcher wants to determine the width of a pond from east to west, which cannot be done by actual measurement. From a point P, he finds the distance to the eastern-most point of the pond to be 8 km, while the distance to the western most point from P to be 6 km. If the angle between the two lines of sight is 60°, find the width of the pond.
Solution :
By finding the missing side, we may find the width of pond.
AP = a, PB = b and AB = c
cos C = (a^{2} + b^{2} - c^{2}) / 2ab
cos 60 = (6^{2} + 8^{2} - c^{2}) / 2(6)(8)
1/2 = (36 + 64- c^{2}) / 96
48 = 100 - c^{2}
c^{2} = 100 - 48
c^{2} = 52
c = √52
= 2 √13 km
So, the width of pond is 2 √13 km.
Problem 2 :
Two Navy helicopters A and B are flying over the Bay of Bengal at same altitude from the sea level to search a missing boat. Pilots of both the helicopters sight the boat at the same time while they are apart 10 km from each other. If the distance of the boat from A is 6 km and if the line segment AB subtends 60° at the boat, find the distance of the boat from B.
Solution :
To find the missing side, we have to use the cosine formula.
cos C = (a^{2} + b^{2} - c^{2})/2ab
Here we use the formula for cos C, because we know the length of b and c.
cos 60 = (a^{2} + 6^{2} - 10^{2})/2(6)(10)
1/2 = (a^{2} -64)/2a(6)
1 = a^{2} - 64/6a
a^{2} - 64 = 6a
a^{2} - 6a - 64 = 0
= (-b ± √b^{2} - 4ac)/2a
= (6 ± √(36 - 4(1)(-64))/2(1)
= (6 ± √292)/2
= (6 ± 2√73)/2
= 3 ± √73
So, the distance from the helicopter B to boat is 3 + √73 km.
Problem 3 :
A straight tunnel is to be made through a mountain. A surveyor observes the two extremities A and B of the tunnel to be built from a point P in front of the mountain. If AP = 3km, BP = 5 km and ∠APB = 120^{◦}, then find the length of the tunnel to be built
Solution :
Given that :
AP = 3 km = b, BP = 5 km = a and ∠APB = 120^{◦}
cos C = (a^{2} + b^{2} - c^{2})/2ab
cos 120 = (5^{2} + 3^{2} - c^{2})/2(5)(3)
cos 120 = cos (90+30) = sin 30 = -1/2
-1/2 = (25 + 9 - c^{2})/30
-15 = 34 - c^{2}
c^{2 } = 49, c = 7
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