Question

1. starting at point a, a ship sails 22.4 km on a course of 174°, then turns and sails 36.5 km on a course of 309°. find the distance from point a to the ship.

Explanation:

Step1: Find internal triangle angle

First, calculate the angle between the two paths inside the triangle formed by point A and the ship's two positions.
The course change from $174^\circ$ to $309^\circ$:
The angle inside the triangle is $180^\circ - (309^\circ - 174^\circ) = 45^\circ$

Step2: Apply the Law of Cosines

Let $b=22.4$ km, $c=36.5$ km, and the included angle $A=45^\circ$. The Law of Cosines states:
$$a^2 = b^2 + c^2 - 2bc\cos(A)$$
Substitute values:
$$a^2 = (22.4)^2 + (36.5)^2 - 2(22.4)(36.5)\cos(45^\circ)$$
Calculate each term:
$(22.4)^2 = 501.76$, $(36.5)^2 = 1332.25$, $\cos(45^\circ)=\frac{\sqrt{2}}{2}\approx0.7071$
$2(22.4)(36.5)(0.7071) = 2(817.6)(0.7071)\approx1156.1$
$$a^2 = 501.76 + 1332.25 - 1156.1 = 677.91$$

Step3: Solve for distance $a$

Take the square root of $a^2$:
$$a = \sqrt{677.91}\approx26.04$$

Answer:

$\approx26.0$ km (rounded to one decimal place)