Peggy drives from Cornwall to Atkins Glen in 46 min. Cornwall is 73 km from Illium in a direction 29° west of south. Atkins Glen is 26.3 km from Illium in a direction 20° south of west. Use Illium as your origin. Find the displacement during the trip in magnitude and direction(north of east).
To find the displacement during the trip, we need to calculate the vectors representing the distances traveled by Peggy from Cornwall to Illium and from Illium to Atkins Glen.
First, let's calculate the vector representing the distance from Cornwall to Illium. We can use the given magnitude (73 km) and direction (29° west of south). To convert this direction into a standard mathematical coordinate system, we can subtract it from 90° (south).
29° - 90° = -61°
Next, we can calculate the x and y components of this vector. Since our origin is Illium, we'll take Illium as the reference point (0, 0).
The x component (horizontal) can be calculated using the cosine of the angle:
x = magnitude * cos(direction) = 73 km * cos(-61°)
The y component (vertical) can be calculated using the sine of the angle:
y = magnitude * sin(direction) = 73 km * sin(-61°)
Now, let's calculate the vector representing the distance from Illium to Atkins Glen. We'll use the given magnitude (26.3 km) and direction (20° south of west). We can convert this direction into a standard mathematical coordinate system by subtracting it from 360° (west).
360° - 20° = 340°
Again, we'll calculate the x and y components of this vector using the cosine and sine functions:
x = magnitude * cos(direction) = 26.3 km * cos(340°)
y = magnitude * sin(direction) = 26.3 km * sin(340°)
Finally, we can calculate the displacement vector by adding the x and y components of the two vectors together:
displacement_x = x(Cornwall to Illium) + x(Illium to Atkins Glen)
displacement_y = y(Cornwall to Illium) + y(Illium to Atkins Glen)
displacement magnitude = sqrt(displacement_x^2 + displacement_y^2)
To determine the direction of the displacement (north of east), we can use the inverse tangent function:
displacement direction = atan2(displacement_y, displacement_x)
By plugging in the values and performing the calculations, you can find the answers to the magnitude and direction of the displacement during Peggy's trip from Cornwall to Atkins Glen.