An athlete ran from town A to town C through town B for 5 hours. If the distance and bearing of town B from town A are 10km and 109 degree respectively; and the distance and bearing of town C from town B are 6km and 205 degree respectively, calculate the speed of the athlete from town A to town B through town C.

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To calculate the speed of the athlete from town A to town B through town C, we need to find the total distance traveled and divide it by the total time taken.

First, let's find the distance from town A to town B through town C. We can use the Law of Cosines to find the side length of the triangle formed by town A, town B, and town C.

Let's denote the side lengths of the triangle as follows:
AB = 10 km (distance from town A to town B)
BC = 6 km (distance from town B to town C)
AC = x km (distance from town A to town C)

We can use the Law of Cosines:

AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(angle ABC)

angle ABC = 180 - angle A - angle B
= 180 - 109 - 205
= -134 degrees

Now, since AC is the side we want to find, we rearrange the equation:

AC = sqrt(AB^2 + BC^2 - 2 * AB * BC * cos(angle ABC))

AC = sqrt(10^2 + 6^2 - 2 * 10 * 6 * cos(-134 degrees))
AC ≈ 7.966 km

Now that we know the distance from town A to town B through town C, we can calculate the total distance traveled:

Total distance = AB + BC = 10 km + 6 km = 16 km

Given that the athlete ran for 5 hours, we can now calculate the speed:

Speed = Total distance / Time taken
Speed = 16 km / 5 hours
Speed ≈ 3.2 km/hour

Therefore, the speed of the athlete from town A to town B through town C is approximately 3.2 km/hour.