Question

1. the first ship leaves port at 1 pm and sail in the direction n34°w at a rate of 20 knots. another ship leaves port at 1:30pm and sails in the direction n56°e at a rate of 18 knots. draw a diagram!

a. how far apart are the ships at 3 pm? (answer in n.m.)

Explanation:

Step1: Calculate time sailed for each ship

First ship: from 1 pm to 3 pm, time = \( 3 - 1 = 2 \) hours. Distance \( d_1 = 20 \times 2 = 40 \) nautical miles (N.M.).
Second ship: from 1:30 pm to 3 pm, time = \( 3 - 1.5 = 1.5 \) hours. Distance \( d_2 = 18 \times 1.5 = 27 \) N.M.

Step2: Determine the angle between their paths

Direction of first ship: N\( 34^\circ \)W, second: N\( 56^\circ \)E. The angle between them: \( 34^\circ + 56^\circ = 90^\circ \). So the triangle formed is right - angled.

Step3: Apply Pythagorean theorem

Let \( d \) be the distance between them. By Pythagoras, \( d = \sqrt{d_1^2 + d_2^2} \).
Substitute \( d_1 = 40 \), \( d_2 = 27 \):
\( d=\sqrt{40^{2}+27^{2}}=\sqrt{1600 + 729}=\sqrt{2329}\approx48.26 \)

Answer:

The ships are approximately \( \boldsymbol{48.26} \) nautical miles apart at 3 pm.