Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane. Yacht B is 500 m due north of yacht A and yacht C is 700 m from A. The bearing of C from A is 015°. (a) Calculate the distance between yacht B and yacht C, in metres to 3 significant figures. (b) The bearing of yacht C from yacht B is θ°, as shown in Figure 1. Calculate the value of θ.

A-Level Edexcel Maths Pure Circles: Figure 1 shows 3 yachts A, B and

Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane - Edexcel - A-Level Maths Pure - Question 8 - 2008 - Paper 2

Question 8

Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane. Yacht B is 500 m due north of yacht A and yacht C is 700 m from A. The beari... show full transcript

Worked Solution & Example Answer:Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane - Edexcel - A-Level Maths Pure - Question 8 - 2008 - Paper 2

Step 1

Calculate the distance between yacht B and yacht C

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Answer

To find the distance between yacht B and yacht C, we can apply the law of cosines. The positions are as follows:

Distance AB = 500 m (north)
Distance AC = 700 m
Angle CAB = 180° - 15° = 165° (since it's a straight line)

Using the law of cosines:

BC^2 = AB^2 + AC^2 - 2 \times AB \times AC \times \cos(CAB)

Substituting the values:

BC^2 = 500^2 + 700^2 - 2 \times 500 \times 700 \times \cos(165°)

Now calculating:

BC^2 = 250000 + 490000 + 2 \times 500 \times 700 \times 0.9659

= 250000 + 490000 + 676715\approx 183515.142

Taking the square root gives:

BC \approx 427.6 m

Thus, rounding to 3 significant figures,

BC = 428 m.

Step 2

Calculate the value of θ

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Answer

To find the bearing of yacht C from yacht B, we need to determine the angle θ. We can use trigonometric relations. For triangle ABC:

AB = 500 m
AC = 700 m
BC = 428 m

Using the sine rule:

\frac{BC}{\sin(A)} = \frac{AC}{\sin(C)}

Where angle C = 15° and angle A = θ. Now substituting the known values:

\frac{428}{\sin(15°)} = \frac{700}{\sin(θ)}

Rearranging gives:

\sin(θ) = \frac{700 \times \sin(15°)}{428}

Calculating this gives:

\sin(θ) \approx 0.773

Now, taking the inverse sine:

θ \approx \sin^{-1}(0.773) \approx 50.5°

The final bearing θ from yacht B to yacht C is approximately 50.5°.

Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane - Edexcel - A-Level Maths Pure - Question 8 - 2008 - Paper 2

Worked Solution & Example Answer:Figure 1 shows 3 yachts A, B and C which are assumed to be in the same horizontal plane - Edexcel - A-Level Maths Pure - Question 8 - 2008 - Paper 2

Calculate the distance between yacht B and yacht C

Calculate the value of θ

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