Cosine Graph Question

Question

The paddle wheel of a ship is 11 feet in diameter, revolves 15 times a minute when moving at top speed, and is 2 feet below the water's surface at its lowest point. Using this speed and starting from a point at the very top of the wheel, write a model for the height, h (in feet), of the end of the paddle relative to the water's surface as a function of time, t (in MINUTES)!

Answer 1

h(t) = A cos(ωt+ø)+h₀

• h(t) = height at time t
• A = amplitude = (1/2)*(11 ft) = 5.5 ft
• ω = angular frequency = 2*pi*(15 revolutions per minute) = 30*pi radians per minute
• t = minutes
• ø = phase shift = 0^o
• h₀ = height of midline above water = 5.5 ft - 2 ft = 3.5 ft

 

h(t) = 5.5*cos(30*pi*t) + 3.5

 

We used the cosine because it's at its max value (top of the wheel) at t = 0.