When a scout in camp. Decided to take a walk for a distance of 42m on a bearing of 032 he than walks for a distance of 28m on a bearing of 154 how far is he from his camp
draw a diagram, figure the angle between the two directions, then use the law of cosines to find the distance.
To find the distance from the scout's camp, we can use the Pythagorean theorem.
Step 1: Draw a diagram to visualize the problem.
Step 2: Calculate the horizontal and vertical components of each leg of the scout's walk using trigonometry.
Leg 1:
Distance = 42m
Bearing = 032
Horizontal component = Distance * sin(Bearing)
Horizontal component = 42m * sin(032°)
Vertical component = Distance * cos(Bearing)
Vertical component = 42m * cos(032°)
Leg 2:
Distance = 28m
Bearing = 154
Horizontal component = Distance * sin(Bearing)
Horizontal component = 28m * sin(154°)
Vertical component = Distance * cos(Bearing)
Vertical component = 28m * cos(154°)
Step 3: Find the total horizontal and vertical components by adding the horizontal and vertical components from each leg.
Total horizontal component = Horizontal component of Leg 1 + Horizontal component of Leg 2
Total vertical component = Vertical component of Leg 1 + Vertical component of Leg 2
Step 4: Use the Pythagorean theorem to find the distance from the camp.
Distance from camp = √((Total horizontal component)^2 + (Total vertical component)^2)
By plugging in the values calculated in the previous steps, we can find the final answer.
To find the distance of the scout from his camp after taking these two walks, we can use the concept of vector addition to calculate their resultant displacement.
First, let's break down the scout's movement:
1. The first walk is a distance of 42m on a bearing of 032. This means that the scout moved 42m in the direction 32 degrees east of north (measured from the north in a clockwise direction).
2. The second walk is a distance of 28m on a bearing of 154. This means that the scout moved 28m in the direction 54 degrees west of south (measured from the south in a clockwise direction).
To calculate the resultant displacement, we can decompose these vectors into their horizontal (north-south) and vertical (east-west) components.
1. For the first walk:
- North component: 42m * sin(32°)
- East component: 42m * cos(32°)
2. For the second walk:
- South component: 28m * sin(54°)
- West component: 28m * cos(54°)
Next, we need to add up these components to get the final north and west values:
- Final north value = (North component of the first walk) - (South component of the second walk)
- Final west value = (East component of the first walk) - (West component of the second walk)
Finally, we can calculate the distance from the camp to the scout using the Pythagorean theorem:
- Distance = sqrt((Final north value)^2 + (Final west value)^2)
Now, let's calculate the final displacement and distance:
1. North component of the first walk:
- 42m * sin(32°) = 22.01m (rounded to two decimal places)
2. East component of the first walk:
- 42m * cos(32°) = 35.79m (rounded to two decimal places)
3. South component of the second walk:
- 28m * sin(54°) = 21.22m (rounded to two decimal places)
4. West component of the second walk:
- 28m * cos(54°) = 14.74m (rounded to two decimal places)
5. Final north value:
- North component of the first walk - South component of the second walk = 22.01m - 21.22m = 0.79m (rounded to two decimal places)
6. Final west value:
- East component of the first walk - West component of the second walk = 35.79m - 14.74m = 21.05m (rounded to two decimal places)
7. Distance from camp to the scout:
- sqrt((Final north value)^2 + (Final west value)^2) = sqrt((0.79m)^2 + (21.05m)^2) ≈ 21.12m (rounded to two decimal places)
Therefore, the scout is approximately 21.12 meters away from the camp.