Question

The blades of a wind turbine are 150 feet long and are attached to the tower at a point 280 feet off the ground. A fly lands on the end of a turbine when it is at its highest point and hangs on through three complete rotations, flying off 90 seconds after it lands. Write a function h(t) to represent the height of the fly t seconds after it lands.

Answer 1

h = 150 cos (2π t / 30) + 280

Explanation:

Vertically, the fly moves up and down in a wave. We can model that wave using:

h = A sin (2π t/T + B) + C

where A is the amplitude of the wave,

T is the period of the wave (time of one revolution),

B is the horizontal phase shift,

and C is the vertical offset.

We can also use cosine instead of sine.

The fly lands on the blade when it is at its highest point. The matches the graph of cosine. So we'll use cosine and make the phase shift 0.

h = A cos (2π t/T) + C

The amplitude of the wave is the length of the blade: 150 ft.

The period is the time for one revolution: 90 s / 3 = 30 s.

The vertical offset is the height of the tower: 280 ft.

h = 150 cos (2π t / 30) + 280

Graph: desmos.com/calculator/jli01ruodt