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A waterwheel, that supplies a mill, has a diameter of 40 feet. You start your stop-watch at a point P, which is on the circumference of the waterwheel and on an axis through the center of the waterwheel, is parallel to the surface of the water. The wheel turns counter-clockwise at a constant rate. Two seconds later, point P on the rim of the wheel is at its greatest height. The center of the waterwheel is located 15 feet above the water and 15 feet to the right of the mill which is our designated frame of reference. You are to model the distance d of point P from the surface of the water in terms of the number of seconds t the stopwatch reads. Draw a picture of the situation labeling all important points and find the sinusoidal equation that models this situation and provide a graph.

Sagot :

A sinusoidal equation can be used to model the height of the

waterwheel above the water.

Response:

  • The sinusoidal function of the waterwheel is; [tex]\underline{d = 20 \cdot sin \left(\dfrac{\pi}{4} \cdot t\right )+ 15}[/tex]
  • A drawing and a graph of the function is attached

Which method can be used to model the situation?

Diameter of the wheel = 40 feet

Point P is a point on an axis parallel to the water surface on the circumference of the wheel.

The time it takes point to reach maximum height = 2 seconds

Location of the center of the water wheel = 15 feet above the center of the water and 15 feet to the right of the mill.

Required:

The model of the distance of the point P from the surface of the water.

Solution:

The distance of the point p above the water surface vary sinusoidally,

according to the following equation;

h = A·sin(ω·t + ∅)) + k

The time it takes the wheel to complete a cycle, T = 4 × 2 s = 8 s

[tex]T = \mathbf{\dfrac{2 \cdot \pi}{\omega}}[/tex]

Therefore;

[tex]\omega = \dfrac{2 \cdot \pi}{8} = \dfrac{\pi}{4}[/tex]

A = The amplitude = The radius of the wheel = [tex]\frac{40 \, ft.}{2}[/tex] = 20 ft.

The vertical shift, k = 15

The horizontal shift is given by the equation;

At t = 0, sin(ω×0 + ∅) = 0

sin(∅) = 0

∅ = 0

The sinusoidal equation that models the distance d is therefore;

  • [tex]\underline{d = 20 \cdot sin \left(\dfrac{\pi}{4} \cdot t\right )+ 15}[/tex]

Please find attached the drawing of the situation and graph of the

sinusoidal equation.

Learn more about sinusoidal equations here:

https://brainly.com/question/12078395

View image Oeerivona
View image Oeerivona
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