Question

a plane took off at a 15° angle to the ground and flew in sydneys direction. she is standing at a point 25 miles from the point where the plane took off. if the plane has traveled 20 miles in the air, about how far is sydney from the airplane? round your answer to the nearest whole number.
8 mi
15 mi
28 mi
59 mi

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Here, \(a = 20\), \(b=25\), and \(C = 15^{\circ}\). First, find \(\cos(15^{\circ})=\cos(45^{\circ}-30^{\circ})=\cos45^{\circ}\cos30^{\circ}+\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659\).

Step2: Substitute values into the formula

Substitute \(a = 20\), \(b = 25\), and \(\cos C\approx0.9659\) into \(c^{2}=a^{2}+b^{2}-2ab\cos C\).

$$
LATEXBLOCK0
$$

Answer:

8 mi