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?
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To find the distance from city C to city A, we can use the concept of vectors.
Step 1: Draw a diagram representing the given information.
Let's assume city B is the origin (0,0) on a coordinate plane. City A is 300km due east, so it would be at the point (300,0). City C is 200km on a bearing of 123° from city B.
Step 2: Determine the coordinates of city C relative to city B.
To do this, we can use the concept of vector components. We'll break down the distance and direction of C into its horizontal and vertical components.
The horizontal component can be found using the formula:
Horizontal Component (Cx) = Distance * cos(Bearing)
Cx = 200 * cos(123°)
The vertical component can be found using the formula:
Vertical Component (Cy) = Distance * sin(Bearing)
Cy = 200 * sin(123°)
Step 3: Find the absolute coordinates of city C.
To find the absolute coordinates of city C, we need to add the coordinates of city C relative to city B to the coordinates of city B.
Absolute X coordinate of C = X coordinate of B + Cx
Absolute Y coordinate of C = Y coordinate of B + Cy
Absolute X coordinate of C = 0 + Cx = Cx
Absolute Y coordinate of C = 0 + Cy = Cy
Step 4: Determine the distance from city C to city A.
Now that we have the coordinates of city C, we can find the distance between city C and city A using the distance formula:
Distance = √((X2 - X1)^2 + (Y2 - Y1)^2)
Distance = √((300 - Cx)^2 + (0 - Cy)^2)
Substituting the values of Cx and Cy:
Distance = √((300 - 200 * cos(123°))^2 + (0 - 200 * sin(123°))^2)
Calculating this expression will give us the distance from city C to city A.
To find the distance from City C to City A, we can use the concept of vectors and the Pythagorean theorem.
Step 1: Draw a diagram representing the given information. Place City B as the starting point and draw a line due east for 300km to City A. Then, from City B, draw a line on a bearing of 123° for 200km to City C.
Step 2: Determine the components of the distance from City B to City C. Since the bearing is given, we can use trigonometry to find the horizontal and vertical components of the line from B to C. The horizontal component can be found by multiplying the distance (200 km) by the cosine of the angle (123°), and the vertical component can be found by multiplying the distance by the sine of the angle.
Horizontal component (BC_h) = 200 km * cos(123°)
Vertical component (BC_v) = 200 km * sin(123°)
Step 3: Calculate the horizontal distance from City A to City C. Since City A is due east of City B, the horizontal distance between City A and City C will be the same as the horizontal component (BC_h).
AC_h = BC_h
Step 4: Calculate the vertical distance from City A to City C. Since City A is due east of City B, there is no vertical component between City A and City C.
AC_v = 0
Step 5: Use the Pythagorean theorem to calculate the distance from City A to City C.
AC = √(AC_h² + AC_v²)
= √(AC_h² + 0)
= √(AC_h²)
= AC_h
Therefore, the distance from City C to City A is equal to the horizontal component, which is BC_h. So, to find the answer, calculate BC_h using the formula:
BC_h = 200 km * cos(123°)