Question

it turns out that the job at the police station is mostly indoors and it interfering with your tan. so, you quit that job and find a job working for tourism perth which offers hot - air balloon rides to tourists. your first assignment is to analyze the and determine the distance separating the balloons.
two hot air balloons approach a landing field. one balloon is 12 meters from the landing point, the other balloon is 17 meters from the same landing point. the angle between the balloons is 70 degrees, how far in the air are the balloons? round your final distance to the nearest meters?

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides (12 m and 17 m) and the included angle (70°) between them. So we use the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( a = 12 \), \( b = 17 \), \( C = 70^\circ \).

Step2: Calculate \( a^2 \), \( b^2 \), and \( 2ab\cos(C) \)

• \( a^2 = 12^2 = 144 \)
• \( b^2 = 17^2 = 289 \)
• \( 2ab = 2\times12\times17 = 408 \)
• \( \cos(70^\circ) \approx 0.3420 \), so \( 2ab\cos(C) \approx 408\times0.3420 \approx 139.54 \)

Step3: Compute \( c^2 \) and \( c \)

• \( c^2 = 144 + 289 - 139.54 = 433 - 139.54 = 293.46 \)
• \( c = \sqrt{293.46} \approx 17 \) (rounded to nearest meter)

Answer:

17 meters