City A is 300km due east of city B. City C is 200km on a bearing of 123¤ from city B. How far is it from C to A?
171km
yes I get 171.346
123-90 = 33 south of east
law of cosines
b^2 = a^2 + c^2 - 2 a c cos B
b^2 = 200^2 + 300^2 - 2(200*300) cos 33
To find the distance from city C to city A, we can use the concept of bearing and distance. Here's how you can solve it step by step:
1. Start by visualizing the given information. Draw a diagram representing the positions of cities A, B, and C. Place city B as the starting point in the center, city A due east (right), and city C on a bearing of 123 degrees (counter-clockwise) from city B.
2. Using the given information, you know that city A is 300km due east of city B. This means that the straight-line distance between city A and city B is 300km.
3. Locate city C on the diagram based on the bearing information. The bearing of 123 degrees indicates an angle of 123 degrees measured from the reference line (east) in a counter-clockwise direction. Measure 200km on this bearing from city B and place city C at that point.
4. Now, you have formed a triangle with sides AB, BC, and AC. We need to find the length of side AC, which represents the distance from city C to city A.
5. Since we have a right-angled triangle, we can use the trigonometric functions to find the missing side. In this case, we can use the sine function because we have the opposite side (AB) and the hypotenuse (BC).
6. Determine the angle between the sides AB and BC using the relationship between bearings and angles. Since the bearing of city C is given as 123 degrees, subtract 90 degrees (the right angle) to find the angle between AB and BC: 123° - 90° = 33°.
7. Apply the sine function: sin(angle) = opposite/hypotenuse. In this case, sin(33°) = AB/BC.
8. Rearrange the equation to solve for AB: AB = sin(33°) * BC.
9. Substitute the known values: AB = sin(33°) * 200km. Calculate the value of sin(33°) using a calculator or mathematical software.
10. Multiply the sine value by 200km to find the length of side AB.
11. The calculated value of AB is the distance from city C to city A.