Question

question 4
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?
19 km, 33° north of west
33 km, 33° north of west
31 km, 57° north of west
24 km, 57° north of west
6.6 km, 40° north of west

Explanation:

Step1: Analyze the right - triangle

Let \(AB = 20\mathrm{km}\) (north - south side) and \(AC=13\mathrm{km}\) (west - east side). The displacement from \(B\) to \(C\) forms the hypotenuse of a right - triangle.

Step2: Calculate the magnitude of displacement

Using the Pythagorean theorem \(d=\sqrt{AB^{2}+AC^{2}}\). Substitute \(AB = 20\mathrm{km}\) and \(AC = 13\mathrm{km}\):

$$d=\sqrt{20^{2}+13^{2}}=\sqrt{400 + 169}=\sqrt{569}\approx24\mathrm{km}$$

Step3: Calculate the direction

Let \(\theta\) be the angle north of west. \(\tan\theta=\frac{AB}{AC}\). Substitute \(AB = 20\mathrm{km}\) and \(AC = 13\mathrm{km}\):
\(\tan\theta=\frac{20}{13}\approx1.54\). Then \(\theta=\arctan(1.54)\approx57^{\circ}\)

Answer:

\(24\mathrm{km},57^{\circ}\) north of west (the fourth option)