A port P is directly east of a port H - Leaving Cert Mathematics - Question 8 - 2013
Question 8
A port P is directly east of a port H.
To sail from H to P, a ship first sails 80 km, in the direction shown in the diagram, to the point R before turning through an... show full transcript
Worked Solution & Example Answer:A port P is directly east of a port H - Leaving Cert Mathematics - Question 8 - 2013
Step 1
Find the distance from R to HP.
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Answer
To find the distance from R to HP, we can use the Law of Sines. First, we need to find angle RHP.
Calculate Angle RHP:
Angle RHP = 180° - (36° + 124°) = 20°.
Now, apply the Law of Sines:
sin(20°)d=sin(124°)110
where d is the distance from R to HP.
Rearranging gives us:
d=110⋅sin(124°)sin(20°)
After calculating, we find:
d≈47.02 km.
Step 2
Calculate |HP|.
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Answer
To find |HP|, we can again use the Law of Cosines:
Identify components:
|HP|² = |HR|² + |RP|² - 2|HR||RP|\cos(124°).
Given |HR| = 80 km and |RP| = 110 km.
Plugging in the values gives us:
∣HP∣2=802+1102−2(80)(110)cos(124°).
Simplifying this, we find:
∣HP∣≈155.15 km.
Step 3
Find |RT|.
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Answer
To find |RT|, we can utilize the relationships established:
Understand the relationships:
Given |HT| = 110 km and |RP| = 80 km, we can calculate:
∣RT∣=∣HT∣−∣HR∣
Compute using calculations from before:
Angle RHP (20°) gives us a triangle where the opposite side can be considered for |RT|. Using trigonometry or further simplifications based on coordinates, find:
∣RT∣≈36.56 km.
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