Final answer:
The student is asked to calculate the distance between Town B and Town C given the bearings and distances from Town A to both towns. To solve this, we use the Law of Cosines with the given sides and the angle found by subtracting the bearings. The distance is the square root of the sum of the squares of the two known sides minus twice their product multiplied by the cosine of the included angle.
Step-by-step explanation:
To calculate the distance between Town B and Town C, we must first understand the concept of bearings and how they can be used to create a triangle with the positions of the towns. Bearings are measured in degrees, starting from the north direction in a clockwise manner. Town B is 73 km away from Town A at a bearing of 069°. Town C is 64 km away from Town A at a bearing of 112°. The angle between the lines from A to B and A to C can be found by subtracting the smaller bearing from the larger one, which in this case is 112° - 069° = 43°.
We now have a triangle formed by the three towns, with two sides and the included angle known. We can solve for the distance between Town B and Town C, which is the third side of the triangle, using the Law of Cosines. The Law of Cosines states that for any triangle with sides a, b, and c and an angle C opposite to side c, the equation is: c² = a² + b² - 2ab*cos(C). Applying this to our triangle, with a = 73 km, b = 64 km, and C = 43°, we get:
c² = (73)^2 + (64)^2 - 2*(73)*(64)*cos(43°)
Solving this equation will give us the distance between Town B and Town C. Make sure to calculate the cosine of 43° in radians or to set your calculator to degree mode.