A car starts from the origin and is driven 1.72 km south, then 3.11 km in a direction 58° north of east. Relative to the origin, what is the car's final location? Express your answer in terms of an angle (in degree)
and a distance.
X = hor. = 3.11cos58 = 1.65km.
Y=ver. = 3.11sin58 + (-1.72) = 0.92km.
tanA = Y / X = 0.92 / 1.65 = 0.5560,
A = 29.1 deg.
R = X / cosA = 1.65 / cos29.1 = 1.89km
@ 29.1 deg. North of East.
To find the car's final location relative to the origin, we can break down the car's motion into two components: one in the north-south direction and the other in the east-west direction.
First, let's find the north-south component. The car travels 1.72 km south, so we can represent this as a negative displacement. Therefore, the north-south component is -1.72 km.
Next, let's find the east-west component. The car travels 3.11 km in a direction 58° north of east. To find the east-west component, we need to multiply the total distance by the cosine of the angle since the cosine function gives the adjacent/hypotenuse ratio in a right triangle. Therefore, the east-west component is 3.11 km * cos(58°).
Now, let's calculate the east-west component:
east-west component = 3.11 km * cos(58°)
= 3.11 km * 0.530
= 1.6493 km
So, the east-west component is approximately 1.6493 km.
Now, to find the car's final location relative to the origin, we need to add the north-south component and the east-west component:
final location = north-south displacement + east-west displacement
= -1.72 km + 1.6493 km
Simplifying, we get:
final location = -0.0707 km
Therefore, the car's final location, relative to the origin, is approximately 0.0707 km west of the origin. To express this as an angle, we can consider the direction from the origin to the final location as a vector. The angle formed between this vector and the positive x-axis can be found using the inverse tangent function:
angle = arctan(north-south component / east-west component)
= arctan((-1.72 km) / (1.6493 km))
Calculating, we find:
angle ≈ -48.8°
Therefore, the car's final location, relative to the origin, is approximately 0.0707 km west of the origin at an angle of -48.8°.