Town A lies 20 km north of town B. Town C lies 13 km west of town A.A small plane flies directly from town B to town C. What is the displacement of the plane ?

Options

A. `33 km, 33 ^(@)` north of west
B. `19 km , 33 ^(@)` north of west
C. `24km, 57^(@)` north of west
D. `31 km, 57^(@)` north of west

Answer

To solve the problem step by step, we will first establish the positions of towns A, B, and C, and then calculate the displacement of the plane flying from town B to town C.

### Step 1: Establish Coordinates
- Let town B be at the origin (0, 0).
- Town A is 20 km north of town B, so its coordinates are (0, 20).
- Town C is 13 km west of town A. Since west is in the negative x-direction, the coordinates of town C will be (-13, 20).

### Step 2: Determine the Displacement Vector
- The displacement vector from town B (0, 0) to town C (-13, 20) can be represented as:
\[ \vec{BC} = \vec{C} - \vec{B} = (-13 - 0, 20 - 0) = (-13, 20) \]

### Step 3: Calculate the Magnitude of the Displacement
- The magnitude of the displacement vector can be calculated using the Pythagorean theorem:
\[ |\vec{BC}| = \sqrt{(-13)^2 + (20)^2} = \sqrt{169 + 400} = \sqrt{569} \approx 23.85 \text{ km} \]

### Step 4: Determine the Direction of the Displacement
- To find the direction of the displacement, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{20}{13} \]
- Therefore, we can find the angle \(\theta\):
\[ \theta = \tan^{-1}\left(\frac{20}{13}\right) \approx 57.0^\circ \]

### Step 5: Finalize the Direction
- The angle \(\theta\) is measured from the west towards the north. Thus, the direction of the displacement can be expressed as "57 degrees north of west".

### Final Answer
The displacement of the plane flying from town B to town C is approximately **23.85 km** at an angle of **57 degrees north of west**.

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