A car starts from the origin and is driven 1.3 km south and then 3.5 km in a direction 53° north of east. Relative to the origin, what is the car's final location? Express your answer in terms of a distance and an angle.
To find the car's final location relative to the origin, we can use vector addition. First, let's break down the car's motion into its components.
The car moves 1.3 km south, so its southern component is -1.3 km.
The car then moves 3.5 km in a direction 53° north of east. To find the eastern component, we can use trigonometry. The angle between the east direction and the north of east direction is 90° - 53° = 37°. So, the eastern component is 3.5 km * cos(37°).
To find the northern component, we can also use trigonometry. The angle between the east direction and the north of east direction is 90° - 37° = 53°. So, the northern component is 3.5 km * sin(53°).
Now, let's add up the components.
The total southern component is -1.3 km.
The total eastern component is 3.5 km * cos(37°).
The total northern component is 3.5 km * sin(53°).
To find the final location, we add the components together:
Final southern component = -1.3 km + 0 km = -1.3 km (since there is no southern component in the east direction)
Final eastern component = 0 km + 3.5 km * cos(37°)
Final northern component = 0 km + 3.5 km * sin(53°)
Now, we can use this information to find the distance and angle of the car's final location relative to the origin.
The distance is given by the Pythagorean theorem:
Distance = sqrt((Final eastern component)^2 + (Final northern component)^2)
The angle can be found using trigonometry:
Angle = arctan(Final northern component / Final eastern component)
Plug in the values and calculate to get the final answer.