An airplane flies 200km due west from city A to city B and then 305km in the direction of 33.5 degrees north of west from city B to city C.

a.) in straight-line distance, how far is city C from city A?

b.) relative to city A,in what direction is city C?

c.)Why is the answer only approximately correct?

To find the answer to these questions, we can use vector addition.

a) To determine the straight-line distance from City A to City C, we need to add the displacement vectors for both legs of the journey. Let's break down the problem step by step:

First, let's determine the components of the displacement vector from City A to City B.

The airplane flies 200 km due west. In terms of components, this translates to a displacement vector of (-200, 0), where the x-component is -200 km (westward direction) and the y-component is 0 km (since there is no displacement in the north-south direction).

Next, let's determine the components of the displacement vector from City B to City C.

The airplane travels 305 km in the direction of 33.5 degrees north of west. To determine the components, we can break down the displacement vector into its x and y components.

The x-component is found by multiplying the magnitude (305 km) by the cosine of the angle (33.5 degrees): x = 305 km * cos(33.5°)

The y-component is found by multiplying the magnitude (305 km) by the sine of the angle (33.5 degrees): y = 305 km * sin(33.5°)

Now, we sum up the components of both displacement vectors to find the total displacement vector from City A to City C:

x_total = -200 km + (305 km * cos(33.5°))
y_total = 0 km + (305 km * sin(33.5°))

Finally, we can calculate the magnitude of the total displacement vector using the Pythagorean theorem:

Distance from City A to City C = sqrt(x_total^2 + y_total^2)

b) The direction of City C relative to City A can be determined using trigonometry. We can calculate the angle between the x-axis (east direction) and the total displacement vector from City A to City C using the inverse tangent function:

Angle = arctan(y_total / x_total)

c) The answer is only approximately correct because we assumed the plane flew straight lines from City A to City B and from City B to City C. In reality, wind conditions, air traffic control, or other factors may cause the plane to deviate from a straight path. Additionally, small errors in the measurements of distances and angles can affect the accuracy of the final result.

a.) To find the straight-line distance from city A to city C, we can use the Pythagorean theorem.

Let's call the distance from city A to city C as 'd'.

According to the given information, the distance from A to B is 200 km, and the distance from B to C is 305 km in the direction of 33.5 degrees north of west.

Using trigonometry, we can find the horizontal distance traveled from B to C:
Horizontal distance = 305 km * cos(33.5 degrees)

Now, using the Pythagorean theorem, we can calculate the straight-line distance 'd':
d^2 = (200 km)^2 + (305 km * cos(33.5 degrees))^2

b.) To find the direction of city C relative to city A, we can use trigonometry again. We need to find the angle between the straight-line distance and the west direction.

Let's call this angle 'θ'.

Tan(θ) = (305 km * sin(33.5 degrees)) / (200 km + 305 km * cos(33.5 degrees))

c.) The answer to part a is only approximately correct because it assumes the motion of the airplane is a straight line. In reality, airplanes follow curved flight paths between cities due to various factors such as air traffic control instructions, wind patterns, and changes in altitude. Therefore, the straight-line distance calculated in part a is an approximation and may not exactly match the actual distance between city A and city C. Additionally, the angle calculated in part b is based on idealized straight-line distances and may not accurately represent the actual direction of city C relative to city A.

484 km W20N

as all these numbers are measurements, they can be accurate, but not exact. Also, there is wind speed, which affects the ground speed of the plane. Also, there is curvature of the earth and uneven terrain.