Two airplanes left a town at the same distance of 2km and 5km on a bearing of 240° and 130° respectively calucalate the distance between the two airplanes.

Easy sketch to make.

You have a triangle of sides 2 and 5 with an angle of (240-130)° or 110°

by the cosine law:
d^2 = 2^2 + 5^2 - 2(2)(5)cos110° , where d is the distance between them
d^2 = .....
d = √....

To calculate the distance between the two airplanes, 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 formula is:

c² = a² + b² - 2ab * cos(C)

In this case, we can consider the distance between the two airplanes as the side of the triangle we want to find, and the distances of the airplanes from the town as the two other sides of the triangle.

Let's calculate the distance between the two airplanes using the given information:

Airplane 1:
Distance from the town (a) = 2 km
Bearing (angle A) = 240°

Airplane 2:
Distance from the town (b) = 5 km
Bearing (angle B) = 130°

Using the law of cosines:

c² = a² + b² - 2ab * cos(C)

c² = (2 km)² + (5 km)² - 2(2 km)(5 km) * cos(130° - 240°)

c² = 4 km² + 25 km² - 20 km² * cos(-110°)

c² = 29 km² + 20 km² * cos(110°)

Since cos(110°) = -cos(70°), we can simplify:

c² = 29 km² + 20 km² * (-cos(70°))

c² = 29 km² - 20 km² * cos(70°)

Calculating cos(70°):

cos(70°) ≈ 0.3420

c² = 29 km² - (20 km² * 0.3420)

c² = 29 km² - 6.84 km²

c² ≈ 22.16 km²

Taking the square root of both sides:

c ≈ sqrt(22.16 km²)
c ≈ 4.707 km

Therefore, the approximate distance between the two airplanes is 4.707 km.

To calculate the distance between the two airplanes, we can use the law of cosines. However, since the given bearings are measured with respect to the North direction (0°), we need to convert them to angles with respect to the East direction (90°).

To convert the bearings, we subtract the given angles from 90°:

For the first airplane:
Bearing from the East = 90° - 240° = -150° (clockwise from East)

For the second airplane:
Bearing from the East = 90° - 130° = -40° (clockwise from East)

Now, using the law of cosines, we have:

c^2 = a^2 + b^2 - 2ab*cos(C)

Where c is the distance between the two airplanes, a is the distance of the first airplane from the origin (2km), b is the distance of the second airplane from the origin (5km), and C is the angle between the two airplanes.

In this case, C will be the difference between the two converted bearings:

C = |(-150°) - (-40°)| = |-150° + 40°| = 110°

Now, we can plug the values into the formula:

c^2 = (2km)^2 + (5km)^2 - 2(2km)(5km) * cos(110°)

c^2 = 4km^2 + 25km^2 - 20km^2 * cos(110°)

c^2 = 29km^2 - 20km^2 * cos(110°)

c^2 ≈ 29km^2 - (-20km^2 * 0.45399) [cos(110°) ≈ 0.45399]

c^2 ≈ 29km^2 + 9.0798km^2

c^2 ≈ 38.0798km^2

Taking the square root of both sides, we get:

c ≈ √(38.0798km^2)

c ≈ 6.17km

Therefore, the distance between the two airplanes is approximately 6.17km.