Question

two boats leave a port at the same time. boat a travels east at a speed of 12 km/h, and boat b travels southwest at a speed of 14 km/h. determine how far apart the boats are after two hours. answer accurate to the nearest kilometre.

Explanation:

Step1: Calculate distance traveled by each boat

Boat A: speed = 12 km/h, time = 2 h, distance \( d_A = 12\times2 = 24 \) km (east).
Boat B: speed = 14 km/h, time = 2 h, distance \( d_B = 14\times2 = 28 \) km (southwest).

Step2: Determine the angle between their paths

East and southwest form a \( 90^\circ + 45^\circ = 135^\circ \) angle (since southwest is 45° from south and west, east is 90° from south).

Step3: Apply the Law of Cosines

Let \( c \) be the distance between them. Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( a = 24 \), \( b = 28 \), \( C = 135^\circ \).
\( \cos(135^\circ) = -\frac{\sqrt{2}}{2} \approx -0.7071 \).
\( c^2 = 24^2 + 28^2 - 2\times24\times28\times(-0.7071) \)
\( c^2 = 576 + 784 + 2\times24\times28\times0.7071 \)
\( c^2 = 1360 + 2\times24\times28\times0.7071 \)
Calculate \( 2\times24\times28 = 1344 \), \( 1344\times0.7071 \approx 950.5 \)
\( c^2 \approx 1360 + 950.5 = 2310.5 \)
\( c \approx \sqrt{2310.5} \approx 48.07 \)

Answer:

48