The parasailing system shown uses a winch to pull the rider in towards the boat, which is traveling with a constant velocity. During the interval when theta is between 20° and 40°, (where t=0 at theta=20°) the angle increases at the constant rate of 2°/s. During this time, the length of the rope is defined by the relationship r=125-(1/3)t^(3/2), where r and t are expressed in meters and seconds, respectively. At the instant when the rope makes a 30-degree angle with the water, the tension in the rope is 18kN. At this instant, what is the magnitude and direction of the force of the parasail on the 75 kg parasailor?

The magnitude of the force is  

the direction is 57.98° from the horizontal plane in a counter clockwise direction

Explanation:

From the question we are told that

      At t = 0 ,

      The rate at which the angle increases is

Converting this to revolution per second  

     The length  of the rope is defined by

    At , The tension on the rope T = 18 kN

      Mass of the para-sailor is

Looking at the question we see that we can also denote the equation by which the length is defied as an an equation that define the linear displacement

  Now the derivative of displacement is velocity

   So

represents the velocity, again the derivative of velocity gives us acceleration

So

Now to the time when the rope made angle of 30° with the water

      generally angular velocity is mathematically represented as

Where is the angular displacement

      Now considering the interval between we have

making t the subject

Now at this time the displacement is

The linear velocity is

The linear acceleration is

Generally radial acceleration is mathematically represented by

Generally angular acceleration  is mathematically represented by

Now

So

The net resultant  acceleration is mathematically represented as

Now the direction of the is acceleration is mathematically represented as

The force on the para-sailor along y-axis is mathematically represented as

The force on the para-sailor along x-axis is mathematically represented as

The net resultant force is mathematically evaluated as

The direction of the force is