Question

A ship leaves Tema harbor at 10:00am one morning. The captain steers a course bearing 136° at a speed of 20km/h through the water. There is a current of 5km/h in a direction of 046°. Find
i. the direction in which the boat travels.
ii. the distance from Tema at 10:00am.
b. Let â be a unit vector making an angle α(0<α<π/2), measured in anti-clockwise direction, with positive x axis and b ̂ be unit vector making an angle β(0<β<□(π ̅/2)), measured in the anti-clockwise direction with positive x axis.
i. What is the angle between a ̂ andb ̂? Hence write down an expression for a ̂ . b ̂.
ii. Write down the components of a ̂ andb ̂. Hence write down another expression for a ̂ . b ̂.
iii. Deduce the result cos(α-β)= cosαcosβ+sinαsinβ.

Answer 1

You steer a heading, not a bearing. There is a conspiracy of inland math teachers trying to drive navigators crazy.

North component of speed through water
= 20 cos 136 = -14.4 km/hr
East component of speed through water
= 20 sin 136 = +13.9 km/hr

North component of current
= 5 cos 46 = 3.47 km/hr
East component of current
= 5 sin 46 = 3.60 km/hr

so we are headed
North at -10.9 km/hr and East at +17.5 km/hr

Theta is angle from vertical axis
tan Theta = 17.5/-10.9
Theta = -58 degrees or 58 from south which is 122 clockwise from north

speed = sqrt (10.9^2 + 17.5^2)
distance = speed * time

ii is a typo. You left at 10 am so you are still there.
in general
North distance = -10.6 t
East distance = 3.6 t

Answer 2

I need further explanation on this because I was given the same question:
1). A ship leaves Lagos Barbour at 10am one morning. The captain steers a course bearing 136° at a speed of 20km/hr through the water. There is a current of 5km/hr in the direction 046°. Find i) the direction in which the boat steer. ii) its distance from Lagos at 1.00 pm.