Question

A Parasail rider wants to reach the maximum height allowed by the Federal Aviation administration... A Parasail rider wants to reach the maximum height allowed by the Federal Aviation administration if the rider’s boat is to tow the parachute at 30 miles per hour. How should the tow to reach the maximum flying height? A. Draw and label a diagram to represent this situation. B. Explain two different methods to solving this problems Method 1: Method 2: C. Find the length of the tow line using both methods from part (b) Write a “ rule of thumb” that does not require trigonometry for calculating the maximum height of a parasail rider given the length of the tow line?

Answer 1

A.) The diagram is below

B.) Method 1: From the diagram, we can see that for the same angle , the height can be increased by using a longer tow line. So, first method to acheive the maximum height is by increasing the length of the tow line.

Method 2 : Again from the diagram, the base of the triangle which is along the sea level depends on the speed of the boat. If the boat covers x distance in 1 sec, then say it is case 1 and if it covers 2x distance in 1 sec, then it might be case2. So second method to achieve the maximum height is by increasing the speed of the boat.

C.) Length of tow line by method 1 :

Given, speed of boat = 30 mi/hr = v = 13.41 m/s . This means in 1 sec, base b = 13.41m (or b = 26.82 m in 2 sec)

suppose = 60 degrees, then Cos 60 = 1/2 = b/l

This implies that l = length of tow line = 2 x b = 2 times speed of the boat in m/s = 2 x 13.41m = 26.82m

Length of tow line by method 2 :

Since speed limit is fixed at 30 miles per hour, so we consider another method. The height can be increased by increasing . Now, when = 90 degrees, height is max. But practically, let us suppose is increased from 60 to only 75 dgrees. Then, Cos 75 = b/l = 13.41/l = 0.2588 ...or l = 13.41/0.2588 = 51.81m

It is noticed that method 2 is more effective than method 1 in increasing the height of the parasailer.

b.) The rule of thumb is that

The length of tow line is directly proportional to the maximum height of the parasail rider. Longer the tow line, greater is the height achieved.