Hal walked from point A to point B due north east from a distance of 20m. He immediately changed direction and walked 50m from B to C on a bearing of 110 degrees.
1. calculate the distance AC
2. Find the bearing of C from A
I made a sketch and got a triangle ABC where AB = 20
BC = 50 and angle B = 70°
by the cosine law:
AC^2 = 20^2+50^2-2(20)(50)cos70
= 2215.959...
AB =47.07 m
Using the sine law, find angle A
To calculate the distance AC, we can use the concept of vector addition.
Step 1: Determine the north and east components of each leg of the journey.
Given that Hal walked from point A to point B due northeast, we can split this movement into its north and east components. Since it is a 45-degree angle from north, both the north and east components of this leg are equal.
North component of AB = 20 * cos(45°)
East component of AB = 20 * sin(45°)
Step 2: Determine the north and east components of the second leg from B to C.
The bearing of 110 degrees means 110 degrees from the north reference point. Therefore, the north and east components can be found as follows:
North component of BC = 50 * sin(110°)
East component of BC = 50 * cos(110°)
Step 3: Add the north and east components to get the total displacement vector.
To find the total displacement vector, we add the north and east components obtained from steps 1 and 2.
Total north component = North component AB + North component BC
Total east component = East component AB + East component BC
Step 4: Calculate the magnitude of the displacement vector using the Pythagorean theorem.
The magnitude of the displacement vector can be calculated using the Pythagorean theorem:
Distance AC = sqrt((Total north component)^2 + (Total east component)^2)
Now let's solve for each component and then calculate the values:
North component AB = 20 * cos(45°) = 20 * 0.7071 ≈ 14.142 m
East component AB = 20 * sin(45°) = 20 * 0.7071 ≈ 14.142 m
North component BC = 50 * sin(110°) = 50 * (-0.9397) ≈ -46.985 m (negative since it is south of the reference point)
East component BC = 50 * cos(110°) = 50 * 0.3420 ≈ 17.100 m
Total north component = 14.142 - 46.985 ≈ -32.843 m
Total east component = 14.142 + 17.100 ≈ 31.242 m
Distance AC = sqrt((-32.843)^2 + (31.242)^2)
= sqrt(1078.72 + 975.48)
= sqrt(2054.20)
≈ 45.3 m
Therefore, the distance AC is approximately 45.3 meters.
To find the bearing of C from A, we can use trigonometry and bearing conventions.
Step 5: Calculate the angle using trigonometry.
The angle θ (bearing of C from A) can be calculated using the arctangent (tan^-1) function:
θ = tan^-1((Total east component) / (Total north component))
Plugging in the values:
θ = tan^-1(31.242 / (-32.843))
The result will be an angle measured counterclockwise from the north direction.
Calculating the angle gives us:
θ ≈ -44.549°
This means that the bearing of C from A is approximately 44.549° in the clockwise direction from the north.