a man prospecting oil in the desert leaves his base camp and drives 42km on a bearing of 032degrees. he then drives 28km on a bearing of 154degrees. how far is he then from his base camp and what is his bearing from it
DRAW IT !!!!
step 1
head 32 deg E of N 42 km
North distance = 42 cos 32 =35.6
East distance = 42 sin 32 = 22.3
step 2
then head 180 - 154 = 26 deg East of South 28 km
North distance = - 28 cos 26 = - 25.2
East distance = 28 sin 26 = 12.3
Total North = 35.6 - 25.2 = 10.4
Total East = 34.6
tan angle East of North = 34.6 /10.4 = 3.33
Angle East of N = 73.3 degrees
distance = sqrt (10.4*2 + 34.6^2)
To find the distance and bearing from the man's current location to his base camp, we can break down his movements into a step-by-step process.
Step 1:
The man drives 42 km on a bearing of 032 degrees from his base camp. This means he moves in a direction of 32 degrees clockwise from the north.
Step 2:
Next, the man drives 28 km on a bearing of 154 degrees. This means he moves in a direction of 154 degrees clockwise from the north.
To find the total distance from the man's current location to his base camp, we can use the Pythagorean theorem because the movements form a right-angled triangle.
Step 3:
Using the Pythagorean theorem, we can calculate the total distance:
Distance = √(42^2 + 28^2)
Distance ≈ √(1764 + 784)
Distance ≈ √(2548)
Distance ≈ 50.48 km
So, the man is approximately 50.48 km away from his base camp.
Step 4:
To find the bearing from the man's current location to his base camp, we can use trigonometry. Using the inverse tangent function, we can find the angle between the horizontal axis (north) and the line connecting the man's current location and his base camp.
Angle (A) = tan^(-1)(28/42)
Angle (A) ≈ 33.69 degrees
Since the bearing is usually given clockwise from the north, we need to subtract Angle (A) from 360 degrees:
Bearing = 360 - 33.69
Bearing ≈ 326.31 degrees
So, the man's bearing from his base camp is approximately 326.31 degrees.
To find the man's distance from his base camp and his bearing from it, we can use trigonometry. Let's break down the problem step by step:
1. Start by drawing a diagram to visualize the situation. Place the base camp on the diagram and mark it as point A.
2. From the base camp, the man drives 42km on a bearing of 032 degrees. To plot this on the diagram, draw a line segment of 42 units (representing kilometers) at a direction angle of 032 degrees from point A. Label the end of this line segment as point B.
3. Next, the man drives 28km on a bearing of 154 degrees. Draw another line segment of 28 units from point B at a direction angle of 154 degrees. Label the end of this line segment as point C.
4. Now, we need to find the length of line segment BC (the distance from point C to point B) and the bearing of point C from point A.
Distance calculation:
- Calculate the horizontal and vertical components of line segment BC using trigonometry. The horizontal component is obtained by multiplying the length of BC (28km) by the cosine of the angle (154 degrees). The vertical component is obtained by multiplying the length of BC by the sine of the angle.
- Use the Pythagorean theorem to find the length of BC. The length of BC is the square root of the sum of the squares of the horizontal and vertical components.
Bearing calculation:
- Find the bearing of point C from point A by calculating the angle between the horizontal axis and line segment BC. This angle can be found using inverse tangent (arctan) function. Make sure to adjust for the appropriate quadrant if necessary.
Now, let's calculate the distance and bearing:
Distance:
- Horizontal component BC = 28 km * cos(154 degrees)
- Vertical component BC = 28 km * sin(154 degrees)
- Length of BC = square root of [(horizontal component BC)^2 + (vertical component BC)^2]
Bearing:
- Angle between the horizontal axis and line segment BC = arctan(vertical component BC / horizontal component BC)
- Bearing of point C from point A = 360 degrees - angle between the horizontal axis and line segment BC (adjust for the quadrant if necessary)
Plug in the values and perform the calculations to find the distance and bearing of point C from point A.