The diagram shows the location of three places X, Y and C. Y is on a bearing of 120° and 15 km from X. C is 40 km from X and lies due west of Y. P lies on the line joining C and Y and is due south of X. a) Find the distance from X to P. b) What is the bearing of C from X, to the nearest degree?

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The diagram shows the location of three places X, Y and C - HSC - SSCE Mathematics Standard - Question 27 - 2023 - Paper 1

Question 27

The diagram shows the location of three places X, Y and C. Y is on a bearing of 120° and 15 km from X. C is 40 km from X and lies due west of Y. P lies on the line j... show full transcript

Worked Solution & Example Answer:The diagram shows the location of three places X, Y and C - HSC - SSCE Mathematics Standard - Question 27 - 2023 - Paper 1

Step 1

Find the distance from X to P.

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Answer

To find the distance from X to P, we first need to determine the angle ∠ZPX in triangle XPY.

Using the fact that the angles in a triangle sum to 180°, we can write:

∠ZPX = 180° - 120° = 60°.

Next, we can apply the cosine rule to find the length of XP:

XP = 15 ext{ km} imes ext{cos}(60°) = 15 ext{ km} imes 0.5 = 7.5 ext{ km}.

Step 2

What is the bearing of C from X, to the nearest degree?

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Answer

Let θ = ∠CXP.

Using the sine rule in triangle CXP, we know:

ext{cos}(θ) = \frac{7.5}{40}.

Calculating this gives:

θ = \text{cos}^{-1}(\frac{7.5}{40}) \approx 79°12'.

Finally, to find the bearing of C from X:

\text{Bearing of C from X} = 180° + 79°12' = 259°12' \approx 259°.\

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