Applications and radial surveys
Solve multi-step problems involving three-figure and compass bearings; find distances and areas using radial (plane-table) surveys; combine the sine rule, cosine rule and area formula in realistic contexts.
Worked examples
Bearings and the cosine rule
Straightforward
Problem
From a lighthouse , boat is 18 km away on a bearing of 045° and boat is 22 km away on a bearing of 105°. Find the distance between the two boats.
1
Find the angle at the lighthouse between and .
2
Apply the cosine rule in triangle , with km, km and included angle .
3
Evaluate.
4
Take the square root to find .
Answer
The distance between the two boats is approximately 20.30 km.
Radial survey — area of a section
Moderate
Problem
In a radial survey from central point , two boundary posts are recorded: m on a bearing of 030° and m on a bearing of 095°. Find the area of triangle .
1
Find the angle at between the two directions.
2
Apply the triangle area formula with the two known sides and included angle.
3
Evaluate.
Answer
The area of triangle is approximately 906 m².
Two-leg journey — finding the interior angle
Challenging
Problem
A drone flies 25 km from base on a bearing of 055° to point . It then flies 30 km on a bearing of 135° to point . Find the straight-line distance from to .
1
Find the interior angle of the triangle at . The drone flew to on bearing 055°, so find the backward direction (from toward ).
Backward direction .
2
Find the interior angle at between the backward direction and the forward direction to (bearing 135°).
3
Apply the cosine rule in triangle with km, km and .
4
Evaluate and the middle term.
, so the term .
5
Substitute and evaluate .
6
Take the square root to find .
Answer
The straight-line distance from to is approximately 42.25 km.
Practise
Q1·Straightforward
A ship sails 40 km on a bearing of 050°. How far north of its starting point is it? Give your answer to 2 decimal places.
Explanation
A bearing of 050° is measured clockwise from north. The northward component is:
Q2·Straightforward
A car travels 30 km on a bearing of 130°. How far east of its starting point is it? Give your answer to 2 decimal places.
Explanation
A bearing of 130° points south-east. The eastward component is:
because is positive in the second quadrant.
because is positive in the second quadrant.
Q3·Straightforward
In a radial survey from point , point is measured on a bearing of 020° and point is measured on a bearing of 085°. What is the angle at point ?
Explanation
Both bearings are measured clockwise from north at .
Q4·Straightforward
In a radial survey from centre point , boundary point is at bearing 035° and boundary point is at bearing 125°. What is the angle ?
Explanation
These two directions are perpendicular to each other.
Q5·Moderate
From port , ship sails 20 km on a bearing of 040° and ship sails 25 km on a bearing of 100°. Find the distance between the two ships to 2 decimal places.
Explanation
The angle .
Applying the cosine rule:
Applying the cosine rule:
Q6·Moderate
In a radial survey from point , two boundary markers are recorded: m on a bearing of 010° and m on a bearing of 080°. Find the straight-line distance to 2 decimal places.
Explanation
Angle at : .
Applying the cosine rule:
Applying the cosine rule:
Q7·Moderate
In a radial survey from point , two boundary points are measured: m at bearing 010° and m at bearing 080°. Find the area of triangle to the nearest square metre.
Explanation
Angle at : .
Q8·Moderate
A helicopter flies 30 km from base on a bearing of 040° to point . It then flies 40 km on a bearing of 100° to point . Find the straight-line distance from base to point , to 2 decimal places.
Explanation
Finding the interior angle at :
- Direction from back to :
- Direction from to :
- Interior angle
Applying the cosine rule in triangle :
- Direction from back to :
- Direction from to :
- Interior angle
Applying the cosine rule in triangle :
Q9·Challenging
A radial survey from point records three boundary points: m at bearing 010°, m at bearing 070°, and m at bearing 140°. Find the total area of the region (the quadrilateral formed by and the three boundary points) to the nearest square metre.
Explanation
Step 1: Angles at .
Step 2: Area of triangle .
Step 3: Area of triangle .
Step 4: Total area.
Step 2: Area of triangle .
Step 3: Area of triangle .
Step 4: Total area.
Q10·Challenging
From point , the bearing to is 058° and m. From , the bearing to is 114°. If m, find the distance to 2 decimal places.
Explanation
The angle at between and is:
Applying the cosine rule:
Applying the cosine rule:
Q11·Challenging
A boat sails from port on a bearing of 040° for 15 km to point . It then turns and sails on a bearing of 110° for 20 km to point . Find the distance to 2 decimal places.
Explanation
Finding the interior angle at :
- Backward direction ( to ):
- Forward direction ( to ):
- Interior angle at :
Applying the cosine rule in triangle :
- Backward direction ( to ):
- Forward direction ( to ):
- Interior angle at :
Applying the cosine rule in triangle :
Q12·Challenging
From lighthouse , ship is 12 km away on a bearing of 035° and ship is 15 km away on a bearing of 110°. Find the distance between the two ships to 2 decimal places.
Explanation
The angle at between and :
Applying the cosine rule:
Applying the cosine rule:
Open Math
Applications and radial surveys
Measurement · MS-M6
Name:
Date:
Q1Straightforward
A ship sails 40 km on a bearing of 050°. How far north of its starting point is it? Give your answer to 2 decimal places.
Q2Straightforward
A car travels 30 km on a bearing of 130°. How far east of its starting point is it? Give your answer to 2 decimal places.
Q3Straightforward
In a radial survey from point , point is measured on a bearing of 020° and point is measured on a bearing of 085°. What is the angle at point ?
Q4Straightforward
In a radial survey from centre point , boundary point is at bearing 035° and boundary point is at bearing 125°. What is the angle ?
Q5Moderate
From port , ship sails 20 km on a bearing of 040° and ship sails 25 km on a bearing of 100°. Find the distance between the two ships to 2 decimal places.
Q6Moderate
In a radial survey from point , two boundary markers are recorded: m on a bearing of 010° and m on a bearing of 080°. Find the straight-line distance to 2 decimal places.
Q7Moderate
In a radial survey from point , two boundary points are measured: m at bearing 010° and m at bearing 080°. Find the area of triangle to the nearest square metre.
Q8Moderate
A helicopter flies 30 km from base on a bearing of 040° to point . It then flies 40 km on a bearing of 100° to point . Find the straight-line distance from base to point , to 2 decimal places.
Q9Challenging
A radial survey from point records three boundary points: m at bearing 010°, m at bearing 070°, and m at bearing 140°. Find the total area of the region (the quadrilateral formed by and the three boundary points) to the nearest square metre.
Q10Challenging
From point , the bearing to is 058° and m. From , the bearing to is 114°. If m, find the distance to 2 decimal places.
Q11Challenging
A boat sails from port on a bearing of 040° for 15 km to point . It then turns and sails on a bearing of 110° for 20 km to point . Find the distance to 2 decimal places.
Q12Challenging
From lighthouse , ship is 12 km away on a bearing of 035° and ship is 15 km away on a bearing of 110°. Find the distance between the two ships to 2 decimal places.
Worked solutions and answers at openmath.au/year-12/standard-2/non-right-angled-trigonometry/applications-and-radial-surveys