A car is driven east for a distance of 46 km, then north for 29 km, and then in a direction 29° east of north for 22 km. Determine (a) the magnitude (in km) of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.

X = 22*cos61 + 46 = 56.67 km.

Y = 22*sin61 + 29 = 48.24 km.

a. D^2 = X^2 + Y^2 = 5538.74
D = 74.4 km.

b. tanA = Y/X = 48.74/56.67 = 0.85124
A = 40.4o.

To determine the car's total displacement, we can break down its motion into east and north components.

(a) Total Displacement:
To calculate the east component of displacement, we use the distance traveled east, which is 46 km.

To calculate the north component of displacement, we use the distance traveled north, which is 29 km.

Since the car also traveled in a direction 29° east of north, we need to find the vector sum of its east and north components.

We can use the trigonometric identities to find the east and north components:

East Component = 22 km * cos(29°)
North Component = 22 km * sin(29°)

To find the total displacement, we need to find the vector sum of the east and north components:

Total Displacement = √(East Component^2 + North Component^2)

(b) Angle of Displacement:
To find the angle of the car's total displacement measured from its starting direction, we can use trigonometry.

We can use the tangent function to find the angle:

Angle of Displacement = arctan(North Component / East Component)

Let's plug in the values:

East Component = 22 km * cos(29°)
North Component = 22 km * sin(29°)

Total Displacement = √(East Component^2 + North Component^2)
Angle of Displacement = arctan(North Component / East Component)

By using these formulas and substituting the given values, we can calculate the magnitude and angle of the car's total displacement from its starting point.

To determine the car's total displacement, we can break down the east and north components of its motion and then combine them to find the resultant displacement.

(a) Magnitude of the car's total displacement:
1. Eastward direction: 46 km
2. Northward direction: 29 km
3. Direction 29° east of north: 22 km

To find the resultant displacement, we can sum up the east and north components using vector addition.

Eastward component (x-axis): 46 km
Northward component (y-axis): 29 km + 22 km * cos(29°)

We can ignore the vertical component of the direction 29° east of north as it does not contribute to the east-west displacement.

Now, let's calculate the values:

Eastward component (x-axis): 46 km
Northward component (y-axis): 29 km + 22 km * cos(29°)

Eastward component (x-axis): 46 km
Northward component (y-axis): 29 km + 22 km * 0.8839
Northward component (y-axis): 29 km + 19.44 km
Northward component (y-axis): 48.44 km

Using the Pythagorean theorem, we can calculate the magnitude of the car's total displacement:

Magnitude = sqrt((Eastward component)^2 + (Northward component)^2)
Magnitude = sqrt((46 km)^2 + (48.44 km)^2)
Magnitude = sqrt(2116 km^2 + 2349.4336 km^2)
Magnitude = sqrt(4465.4336 km^2)
Magnitude ≈ 66.85 km

Therefore, the magnitude of the car's total displacement from its starting point is approximately 66.85 km.

(b) Angle of the car's total displacement measured from its starting direction:
To find the angle, we can use trigonometry:

Angle = arctan((Northward component)/(Eastward component))
Angle = arctan((48.44 km)/(46 km))

Angle ≈ 47.1°

Therefore, the angle (from east) of the car's total displacement measured from its starting direction is approximately 47.1°.