A plane flies from base camp to Lake A, 260 km away in the direction 20.0° north of east. After dropping off supplies it flies to Lake B, which is 150 km at 30.0° west of north from Lake A. Graphically determine the distance and direction from Lake B to the base camp.
Distance ? km
Direction ? °
step pls
To determine the distance and direction from Lake B to the base camp, we can use vector addition and graphical methods.
Step 1: Draw a diagram:
Start by drawing a coordinate system, with North as the positive y-axis and East as the positive x-axis. Label the base camp as point O, Lake A as point A, and Lake B as point B. Mark the distances and angles given in the problem on the diagram.
Step 2: Determine the position vector of Lake A:
Given that the plane flies 260 km in the direction 20.0° north of east, we can break down this vector into its x and y components. The x-component (Ax) can be calculated as Ax = 260 km * cos(20.0°), and the y-component (Ay) can be calculated as Ay = 260 km * sin(20.0°).
Step 3: Determine the position vector of Lake B:
Given that Lake B is 150 km away at 30.0° West of North from Lake A, we can break down this vector into its x and y components relative to Lake A. The x-component (Bx) can be calculated as Bx = 150 km * sin(30.0°), and the y-component (By) can be calculated as By = 150 km * cos(30.0°).
Step 4: Add the position vectors:
Add the position vector of Lake A (Ax, Ay) and the position vector of Lake B relative to Lake A (Bx, By). The resultant vector is the position vector of Lake B relative to the base camp, let's call it vector OB.
Step 5: Find the magnitude and direction:
Calculate the magnitude of the resultant vector OB using the Pythagorean theorem: Magnitude OB = sqrt((OBx)^2 + (OBy)^2).
The direction of the resultant vector OB can be found by taking the inverse tangent of the components OBx and OBy : Direction θ = atan(OBy / OBx).
Step 6: Round the results:
Round the magnitude and direction to the appropriate number of significant figures or decimal places, depending on the problem's requirements.
Remember to consider any negative signs when determining the direction, as they indicate the direction relative to the positive x-axis.