Question

1. two airplanes leave an airport, and the angle between their flight paths is 40°. an hour later, one plane has traveled 300 miles while the other has traveled 200 miles. how far apart are the planes at this time?

Explanation:

Step1: Identify triangle sides and angle

Two sides: 300 miles, 200 miles; included angle: 40°. Use Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$

Step2: Substitute values into formula

$c^2 = 300^2 + 200^2 - 2(300)(200)\cos(40^\circ)$

Step3: Calculate each term

$300^2=90000$, $200^2=40000$, $2(300)(200)=120000$, $\cos(40^\circ)\approx0.7660$

Step4: Compute $c^2$

$c^2 = 90000 + 40000 - 120000(0.7660) = 130000 - 91920 = 38080$

Step5: Find $c$

$c = \sqrt{38080} \approx 195.1$

Answer:

Approximately 195 miles (or 195.1 miles for more precision)