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?
see above
Abc will 33 degreese
To find the distance from city C to city A, we can use the law of cosines. The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. In this case, we can use it to find the distance between C and A, which is the side opposite the angle at city B.
We have the following information:
- City A is 300km due east of city B.
- City C is 200km on a bearing of 123° from city B.
To apply the law of cosines, we need to find the lengths of the sides of the triangle. Let's call the distance from B to C "a", the distance from B to A "b", and the angle at B "C". We can use the given information to find these values.
1. Using the distance formula, we can determine the lengths of sides a and b:
- a = 200km (distance from B to C)
- b = 300km (distance from B to A)
Now, we need to find the measure of angle C.
2. To determine the angle at city B, we can use the given bearing of 123° from city B to city C. Bearing is measured clockwise from north, so we subtract 90° from it to get the angle at city B:
- Angle at B = 123° - 90° = 33°
Now that we have the lengths of the sides and the measure of angle C, we can apply the law of cosines:
c² = a² + b² - 2ab * cos(C)
3. Substitute the known values into the equation:
c² = 200² + 300² - 2 * 200 * 300 * cos(33°)
4. Calculate the value of c² using the cosine function on a scientific calculator or online calculator.
5. Once you have the value for c², take the square root to find c (the distance from C to A).
By following these steps, you can find the distance from city C to city A.