Question

Kelly is standing in a wavy water and notices the depth of the waves varies in a periodic way that can be modeled by a trigonometric function. She starts a stopwatch to time the waves. After 3.2 seconds, and then again every 3 seconds, the water just touches her knees. Between peaks, the water recedes to her ankles. Kelly's ankles are 11cm off the ocean floor, and her knees are 55 cm off the ocean floor. Find the fomula of the trigonometric function that models the depth D of the water t seconds after Kelly starts the stopwatch. Define the function using radians.

Answer 1

A trig wave with average height 0 has the form f(t)=Acos(ωt+ϕ). You could also use the sin - it doesn't matter.

A trig wave with average height b has the form f(t)=b+Acos(ωt+ϕ)

The difference between the max and min heights is 2A. In your problem 2A = 55-12 = 42 so that A = 21.5.

The average wave height is b = 12 + A = 33.5

The time period of the wave is 3, so that its frequency (waves per second) is f = 1/3. Its angular frequency ω (waves per 2π) is 2πf=2π/3.

your wave is now f(t)=33.5+21.5cos(2πt/3+ϕ)

When t = 1.1 the max height is reached so that 55=33.5+21.5cos(2π(1.1)/3+ϕ). Then 1=cos(2π(1.1)/3+ϕ) which in turn means that 2π(1.1)/3+ϕ=0 and solving gives ϕ.

Key Points. One radian is the measure of the central angle of a circle such that the length of the arc is equal to the radius. of the circle. A full revolution of a circle (360∘ ) equals 2π radians 2 π r a d i a n s . This means that 1 radian=180∘π 1 radian = 180 ∘ π

Hope it helps