A fish is reeled in at a rate of 1.9 feet per second from a point 10 feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out?

9514 1404 393

  1.9 degrees per second

Step-by-step explanation:

The angle α the line makes with the water can be described by ...

  Sin = Opposite/Hypotenuse

  sin(α) = 10/(25 -1.9t)

Taking derivatives gives ...

  α'·cos(α) = -10(-1.9)/(25 -1.9t)²

Then the rate of change of the angle at t=0 is ...

  α' = 19/(25²·cos(α))

Of course, cos(α) = √(1 -sin(α)²) = √(1 -(10/25)²) = √(21/25), so we have ...

  α' = 19/(125√21) . . . radians/second

  α' ≈ 0.033269 radians/s ≈ 1.90 degrees/s