Answer:
a) Hence the equation of the sinusoidal function that describes the height of the shorts in terms of time is [tex]y = 9 + 7* sin(\pi * t / 15 - \pi / 6)[/tex]
b) Hence the height of the shorts at exactly t = 10 minutes, to
the nearest tenth of a meter is 5.5 meters
Step-by-step explanation:
a) The wind turbine blade traverses a circular path as it rotates with time (t), whose time variation is given by the following trajectory equation :
[tex]x^2 + (y-yc)^2 = R^2[/tex] ,
where Â
R = (16 m - 2 m)/2 (since diameter = maximum height - minimum height of the pink short)
= 14 m / 2
= 7 m (radius of the circle)
Also, center of the circle will be at (0, 2 + R) i.e (0,9)
So, Â is the trajectory path equation to the circle
Let [tex]x = 7* cos(w*t + \phi ) & y = 9 + 7* sin(w* t + \phi)[/tex] be the parametric form of the above circle equation which represent the position of the pink shorts at the tip of the blade at time t Â
At t= 10s, y = 16 m so we have,
[tex]9 + 7 * sin(10* w + \phi) = 16[/tex] ---------------(1)
Also, at t= 25s, y =2 m so we have,
[tex]9 + 7* sin(25 * w +\phi) = 2[/tex]--------------(2)
Solving we have, [tex]10* w + \phi = \pi/2 & 25*w + \phi = 3*pi/2[/tex]
[tex]15* w = \pi\\\\w = \pi/15 & \phi = \pi/2 - 10*\pi/15 = -\pi / 6[/tex] Â
Therefore [tex]y = 9 + 7* sin(\pi * t / 15 - \pi / 6)[/tex] is the instantaneous height of the pink short at time t ( in seconds)
b) At t= 10minutes = 10 * 60 s = 600s, we have,
[tex]y = 9 + 7 * sin(\pi * 600/15 - \pi / 6)\\\\= 9 + 7 * sin(40* \pi - \pi / 6)[/tex]
= 5.5 meters (pink short will be at 5.5 meters above ground level at t= 10 minutes)
