Question

A bushwalker walks 14 km east and then 9 km south. Find the bearing of his finishing position from his starting point.

Answer 1

Explanation:

To find the bearing of the bushwalker's finishing position from his starting point, we can use trigonometry and the concept of bearings.

First, we can draw a diagram to visualize the bushwalker's journey:

(9 km south)

|

|

(Starting point)

O-----|-------------------> (14 km east)

|

|

(Finishing point)

Next, we can use the tangent function to find the angle between the bushwalker's starting point and finishing point:

tan θ = opposite / adjacent

In this case, the opposite side is 14 km (the distance traveled east), and the adjacent side is 9 km (the distance traveled south):

tan θ = 14 / 9

We can use a calculator or reference table to find that the angle θ is approximately 56.31 degrees.

However, this is not the bearing we are looking for. In the context of bearings, the bearing of a point is the angle measured clockwise from north to the line connecting the starting point and the point in question.

To find the bearing of the bushwalker's finishing point, we need to adjust the angle θ to take into account the fact that bearings are measured from north.

First, we can find the direction of the line connecting the starting point and finishing point. This line travels 14 km east and 9 km south, so it has a slope of -9/14. We can find the angle this line makes with the horizontal axis by taking the arctangent of the slope:

tan α = -9/14

α = -30.96 degrees

Note that we use a negative sign because the line slopes downwards (southward) from left to right.

Finally, we can add this angle to the angle θ we found earlier:

Bearing = 360 - (θ + 90 + α)

= 360 - (56.31 + 90 + (-30.96))

= 287.7 degrees

Therefore, the bushwalker's finishing position has a bearing of approximately 287.7 degrees from his starting point.

Answer 2

`\bf{Answer }`

We can use trigonometry to find the bearing of the finishing position from the starting point.

Using the Pythagorean theorem, we can find the length of the hypotenuse:

c² = a² + b²

where c is the length of the hypotenuse, a is the distance traveled east (14 km), and b is the distance traveled south (9 km).

Substituting the given values, we get:

c² = (14 km)² + (9 km)²

c² = 196 km² + 81 km²

c² = 277 km²

c ≈ 16.67 km (rounded to two decimal places)

Now, we can use trigonometry to find the angle between the hypotenuse and the east direction.

tan(θ) = opposite/adjacent

where θ is the angle between the hypotenuse and the east direction, opposite is the distance traveled south (9 km), and adjacent is the distance traveled east (14 km).

Substituting the given values, we get:

tan(θ) = 9 km / 14 km

θ ≈ 32.47° (rounded to two decimal places)

Therefore, the bearing of the finishing position from the starting point is approximately 32.47° south of east. Alternatively, we can describe the bearing as 157.53° east of south (180° - 32.47°), or simply as southeast.