Question

A bird is diving for fish in the ocean. His height above the water varies sinusoidally with time at 4 seconds, he spots a fish from a maximum height of 112 ft above water. He dives and at 7 seconds, he is at a minimum height of 14ft under water. Write an equation of the bird's height above the water as a function of time.

Answer 1

The equation of the bird's height above the water as a function of time can be expressed as:

H(t) = A * sin (B * t + C) + D

Where:

A is the amplitude, which is the difference between the maximum and minimum

B is angular frequency (2πf)

C is the phase shift

D is the midline

The maximum height is 112 ft and the minimum height is 14 ft, so A = 98ft.

The frequency of the cycle is 4 seconds (1 cycle every 4 seconds).

Therefore, the angular frequency is 2π/4 = π/2

The bird was at maximum height of 112 ft at t=4s, so the phase shift C = 0.

The midline is the average of the maximum and minimum, so D = (112+14)/2 = 63 ft.

Therefore, the equation of the bird's height above the water as a function of time is:

H(t) = 98 * sin (π/2 * t + 0) + 63