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22 A, B, C and D are four towns - OCR - GCSE Maths - Question 21 - 2017 - Paper 1
Question 21
22 A, B, C and D are four towns. B is 25 kilometres due East of A. C is 25 kilometres due North of A. D is 45 kilometres due South of A. (a) Work out the bearing of ... show full transcript
Worked Solution & Example Answer:22 A, B, C and D are four towns - OCR - GCSE Maths - Question 21 - 2017 - Paper 1
Step 1
Work out the bearing of B from C.
Answer
To determine the bearing of B from C, we start by identifying the locations based on the provided details.
- Position A: Assume point A is at the origin (0, 0).
- Position B: Since B is 25 km due East of A, its coordinates are (25, 0).
- Position C: C is located 25 km due North of A, so its coordinates are (0, 25).
Next, we calculate the angle from point C to point B. The bearing is measured clockwise from the North:
- The direction from C to B forms a right triangle with:
- The eastward distance = 25 km (from A to B)
- The northward distance = 25 km (from A to C)
Using trigonometric functions, we need the tangent of the angle θ.
Thus, θ = 45°.
Now, to find the bearing from C to B:
- The bearing from North is calculated as:
So, the bearing of B from C is 045°.
Step 2
Calculate the bearing of D from B.
Answer
To find the bearing of D from B, we first identify the positions:
- Position D: D is located 45 km due South of A, making its coordinates (0, -45).
- Position B: We've established B's position earlier as (25, 0).
Now, we need to determine the direction from B to D:
- The vector from B to D can be calculated as follows:
- The eastward difference = 0 - 25 = -25 km (Westward)
- The southward difference = -45 - 0 = -45 km (Southward)
We now compute the angle from North:
- First, we find the angle θ relative to the east-west line using the tangent function:
Using inverse tangent:
- θ = arctan(1.8) which gives us approximately 61.93°.
Since we are measuring clockwise from North and the direction is in the fourth quadrant (due South and West), we calculate the bearing:
- The bearing from North is:
Thus, we round up the bearing to give the final result as 242°.