Answer:
The light house is moving at the speed of 83.8Km/minute.
Step-by-step explanation:
The question forms a right angle triangle where point O to P is 3km.and P to Q is 1km . The angle between point O and Q which is the light house is thetha.
The lighthousevmakes 4rev/min
dtheta/dt = 4× 2pi/1min.= 8piRad/min
From the right angle ,tan theta = x/3
Differenting both sides with respect to t
d(tantheta)/dtheta = 1/3×dx/dt
d(tantheta)/dtheta × dtheta/ dt = 1/3×dx/dt
Using chain rule
Sec^2theta× dtheta/dt = 1/3×dx/dt
(1 + tan^2 theta)×dtheta/dt = 1/3×dx/dt
When x = 1, we have tan theta= 1/3
(1 + 1/9)×dtheta/dt = 1/3×dx/dt
Substituting dthata/dt = 8 pi
10/9 × 8pi = 1/3×dx/dt
10/3 × 8pi = dx/dt
80pi/3 = dx/dt
dx/dt = (80 × 3.142)/3
dx/dt = 251.36/ 3 = 83.79km/hr
Therefore the lighthouse is moving at a speed of 83.8km/minute
