A surveyor sights two signs and the angle between the two lines of sight measures 55. If the first and the second signs are 70 m and 50 m away,respectively,from the surveyor, find the distance between the two signs .Express your answer to the nearest meter.
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a surveyor sights two signs and the angle between the two lines of sight measures 55 degrees. If the first and the second signs are 70 m and 50 m away, respectively, from the surveyor, find the distance between the two signs. Express your answer to the nearest meter.
plug in the law of cosines:
c^2 = a^2 + b^2 - 2ab cosθ
82 cm
83
To find the distance between the two signs, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following relationship holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we can label a as the distance to the first sign (70 m), b as the distance to the second sign (50 m), and C as the angle between the two lines of sight (55 degrees). We're trying to find the distance between the two signs, which we can label as c.
Plugging in the values into the Law of Cosines equation, we get:
c^2 = (70)^2 + (50)^2 - 2(70)(50) * cos(55)
To calculate this, we can follow these steps:
1. Calculate (70)^2 and (50)^2.
(70)^2 = 4900
(50)^2 = 2500
2. Calculate 2(70)(50) * cos(55).
2(70)(50) * cos(55) ≈ 7000 * 0.5736 ≈ 4015.2
3. Substitute the values back into the Law of Cosines equation.
c^2 = 4900 + 2500 - 4015.2
c^2 = 3384.8
4. Take the square root of both sides to find c.
c ≈ sqrt(3384.8) ≈ 58.2
Therefore, the distance between the two signs is approximately 58 meters.