Question

question 11
a recent spike of bigfoot sightings in the adirondacks has the community clamoring for action, and a team of forest rangers are tracking the creature(s) for possible deportation to canada. the rangers are searching a wide area surrounding their control station, and mitchell is coordinating the operation from lookout peak (see diagram below).
from his vantage point at lookout peak, mitchell spots a flare roughly 4 km away at a bearing of e 55°s. lookout peak is located 6 km from the control station, at a bearing of n 31°e, and mitchell measures an arc of 66° between the flare and the control station. which of the following expressions gives an estimate for the distance between the flare and the control station?

lookout peak
e
66
55
4 km
n
6 km
flare
31
?
control station

√(6² + 4² + 2(6)(4)cos 66)
√(6² + 4² - 2(6)(4)cos 66)
6² + 4² - 2(6)(4)cos 55
√(6² + 4² - 2(6)(4)cos 55)
6cos 59 + 4cos 55

Explanation:

Step1: Identify the triangle - sides and angle

We have a triangle with two - side lengths \(a = 6\) km and \(b = 4\) km and the included angle \(\theta=66^{\circ}\). We want to find the length of the third side \(c\) (distance between the flare and the control station).

Step2: Apply the Law of Cosines

The Law of Cosines formula for finding the length of a side \(c\) of a triangle with sides \(a\) and \(b\) and included angle \(\theta\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Here, \(a = 6\), \(b = 4\), and \(\theta = 66^{\circ}\), so \(c=\sqrt{6^{2}+4^{2}-2(6)(4)\cos66^{\circ}}\).

Answer:

\(\sqrt{6^{2}+4^{2}-2(6)(4)\cos66^{\circ}}\)