Betty Lou was sitting on the front porch of her plantation when the riverboat went by. As the paddlewheel turned, a point on the paddle blade moved in such a way that its distance, d from the...

Betty Lou was sitting on the front porch of her plantation when the riverboat went by. As the paddlewheel turned, a point on the paddle blade moved in such a way that its distance, d from the water's surface was a sinusoidal function of time. Four seconds later, the point was at its highest, 16 feet above the water's surface. The diameter of the wheel was 18 feet, and it completed a revolution every 10 seconds. 

What is the equation of the sinusoid

How far above the surface was the point after 5 seconds, 17 seconds? 

please provide detailed explanation. Thank you!

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ishpiro | College Teacher | (Level 1) Educator

Posted on

The general equation of the sine function is

d = Asin(b(t -c)) +h

Here, A is amplitude, `(2pi)/b` is the period, c is the phase shift and h is the vertical shift.

Since the diameter of the wheel is 18 feet, the amplitude is A = 18/2 = 9. Since the highest point is 16 feet above water surface is 16 feet,  the vertical shift is determined by d = 16 = A + h = 9 +h.

h = 16 - 9 =7.

Since the wheel completes a revolution every 10 seconds, the period of the sine function is 10. Then,

`(2pi)/b = 10` and `b=(2pi)/10 = pi/5`

Finally, since it took 4 seconds to reach 16 ft, we can find c:

`16 = 9*sin(pi/5(4 - c)) + 7`

`sin(pi/5)(4-c) = 1`

`pi/5(4-c) = pi/2`

`4-c = 2.5` and c = 1.5

So the equation of sine wave will be

`d=9sin(pi/5(t-1.5)) + 7`

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