Answer:
The rate of change of the tracking angle is 0.05599 rad/sec
Step-by-step explanation:
Here the ship is traveling at 15 mi/hr north east and
Port to Radar station = 2 miles
Distance traveled by the ship in 30 minutes = 0.5 * 15 = 7.5 miles
Therefore the ship, port and radar makes a triangle with sides
2, 7.5 and x
The value of x is gotten from cosine rule as follows
x² = 2² + 7.5² - 2*2*7.5*cos(45) = 39.04
x = 6.25 miles
By sine rule we have
[tex]\frac{sin A}{a} = \frac{sin B}{b}[/tex]
Therefore,
[tex]\frac{sin 45}{6.25} = \frac{sin \alpha }{7.5}[/tex]
α = Angle between radar and ship α
∴ α = 58.052
Where we put
[tex]\frac{sin 45}{6.25} = \frac{sin \alpha }{x}[/tex] to get
[tex]\frac{x}{6.25} = \frac{sin \alpha }{sin 45}[/tex] and differentiate to get
[tex]\frac{\frac{dx}{dt} }{6.25} = cos\alpha\frac{\frac{d\alpha }{dt} }{sin 45}[/tex]
[tex]\frac{15sin45 }{6.25cos\alpha } =\frac{d\alpha }{dt} }[/tex]= 3.208 degrees/second = 0.05599 rad/sec.
