Aship is approaching a lighthouse the is known to be 30 meters in height. the apparent size of the lighthouse is .005 radians. ten minutes later, the lighthouse has an apparent size of .1 radians. how fast is the ship traveling.

Refer to the diagram shown below.

Let the ship approach the lighthouse at a constant speed of v km/h.

When the angle of elevation is  0.005 radians,
tan(0.005) = 30/(x + y)
x + y = 30/tan(0.005) = 6000 m        (1)

When the angle of elevation is 0.1 radian,
tan(0.1) = 30/x
x = 30/tan(0.1) = 299 m

From (1), obtain
y = 6000 - 299 = 5701 m = 5.701 km

The ship travels 5701 m  in 10 minutes. It's speed is
v = (5.701 km)/(1/6 h) = 34.2 km/h

The ship is traveling at  34.2 km/h.