A swimmer on a floating air mattress will follow the rise and fall of the waves in lake. When the person is on the top of the wave, her feet are 0.5m above the average surface height of the lake. When she is in a trough her feet dig 0.5m into the water (-0.5m). The peaks of the waves are separated by two seconds. Create an equation of the sinusoidal function that models this movement assuming that she starts at the average surface height of the lake and heads upwards.

Quick answer:

A model for the swimmers height in the water at time t is h(t)=.5sin(pi * t).

Expert Answers

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A sinusoid can be modeled with a sine curve with possible transformations. A general formula is `h(t)=asin(b(x-h))+k` where h(t) represents the height above 0 at time t, a is the amplitude of the wave, b affects the period of the wave, h creates a phase shift, and k describes the midline.

The amplitude is the distance (so nonnegative) from the peak (or trough) of the wave to the midline. The coefficient a is that distance. (If a<0 then the graph is a reflection across the midline.) Here that distance is .5m so a=.5

The period is derived from the frequency. The frequency is how many repetitions of the wave occur per unit of time, or cycles per unit of time. The period is the reciprocal of the frequency. The coefficient b is found by taking the typical period of the function (for sine or cosine this is `2pi` ) divided by p. We are told that the wave repeats every 2 seconds so the period is 2 and `b=(2pi)/2=pi` . (If b<0 then the graph is reflected over a vertical line.)

If the wave does not start at the midline, and we are using sine, then there is a phase shift h to contend with. Here we are assuming that the wave begins at the average height (midline) so h=0.

The midline describes the average height. We are not given the average depth of the water when it is still, so we can let k=0. The model we use will describe the motion as a function of how high above or below the swimmer is with respect to the average.

So one possible model is `h(t)=.5sin(pi(t-0))+0=.5sin(pi t)`

Other models include using the cosine with a phase shift of 1/2 to the right or h(t)=.5cos(pi(t-1/2)) or a cosine reflected over the midline after a phase shift of 1/2 to the left: h(t)=-.5cos(pi(t+1/2)) or a sine curve reflected over the midline with a phase shift of 1 to the right, and so on.

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