Question

A contestant on a game show spins a wheel that is located on a plane perpendicular to the floor. He grabs the only red peg on the circumference of the wheel, which is 1.5 m above the floor (the peg is) and pushes it downward. The peg reaches a min height of 0.25 m and a max height of 2.75 m above the floor. Sketch two cycles of the graph representing the height above the ground as a function of the total distance moved. Then determine the equation of the sine function

Answer 1

The spin of the wheel by the contestant is an illustration of a sine function whose equation is $\mathbf{y =-1.25 sin(0.8x) + 0.25}$

The wheel is 1.5 m above the floor, and the peg reaches a maximum height of 2.75 m.

So, the amplitude is:

$\mathbf{A = 1.5 - 2.75}$

$\mathbf{A = -1.25}$

The vertical shift is the minimum height of the peg.

So, we have:

$\mathbf{D = 0.25}$

The period (T) equals one circumference.

So, we have:

$\mathbf{T = (max-min)\pi}$

This gives

$\mathbf{T = (2.75-0.25)\pi}$

$\mathbf{T = 2.5\pi}$

The frequency (B) is then calculated as:

$\mathbf{B = \frac{2\pi}{T}}$

This gives

$\mathbf{B = \frac{2\pi}{2.5\pi}}$

Divide

$\mathbf{B = 0.8}$

A sine function is represented as:

$\mathbf{y =A sin(B(x + C)) + D}$

Substitute the calculated values

$\mathbf{y =-1.25 sin(0.8(x + C)) + 0.25}$

The horizontal shift (C) is 0.

So, the function becomes

$\mathbf{y =-1.25 sin(0.8x) + 0.25}$

See attachment for the graph of the equation

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