Answer to Question #236270 in Math for luka

Question

A boat sails from port A to port B which is 50 km away on a bearing of 126 degrees from A. It then sails to C which is 46 km away on a bearing of 170degrees from B. The boat continues to sail on a course of 045degrees from C to D for 65 km.

(a) How far south of A is D? (4marks)

(b) How far east of A is D? (4marks)

(c) Calculate the bearing of D from A. (3marks)

(first draw the figure indicating the given data.) (4marks)

Answer 1

$A(0,0)$

$B(50\cos(36\degree), -50\sin(36\degree))$

$C(50\cos(36\degree)+46\cos(80\degree),$

$-50\sin(36\degree)-46\sin(80\degree))$

$D(50\cos(36\degree)+46\cos(80\degree)+65\cos(45\degree),$

$-50\sin(36\degree)-46\sin(80\degree)+65\sin(45\degree))$

(a)

$$50\cos(36\degree)+46\cos(80\degree)+65\cos(45\degree)-0$$

$$=50\sin(36\degree)+46\sin(80\degree)-65\sin(45\degree)$$

$$\approx28.728(km)$$

(b)

$$0-(-50\sin(36\degree)-46\sin(80\degree)+65\sin(45\degree))$$

$$=50\cos(36\degree)+46\cos(80\degree)+65\cos(45\degree)$$

$$\approx94.401(km)$$

(c)

$$\theta=180\degree-\tan^{-1}(\dfrac{|x_D|}{|y_D|})$$

$$=180\degree-\tan^{-1}(\dfrac{50\cos(36\degree)+46\cos(80\degree)+65\cos(45\degree)}{50\sin(36\degree)+46\sin(80\degree)-65\sin(45\degree)})$$

$$\approx106.9\degree$$

The bearing of D from A is $106.9\degree.$