The question relates to application of sine functions.

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A carnival Ferris wheel with a radius of 7m makes one complete revolution every 16 seconds. The bottom of the wheel is 1.5m above the ground.

a) Draw a sketch to show a person's height (h) above the ground varies with time (t) for two revolutions, starting when the person gets onto the Ferris wheel at it lowest point.

b) Find the sine equation for the height (h) of the graph with respect to the time (t). (In radians)

--- Do I need to find the diameter for the circle in order to find the equation? And if I do, what do I do next? I don't know how to solve this problem, please help.

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Answers:

b) y = 7sin pi/8 (t-4) + 8.53

You will need the radius r of the wheel to do this problem. Let A be the angle the wheel has rotated from the person's lowest point. The altitude h of the person is 1.5 + r - r cos A

h = 1.5 + 7 (1 - cos A)
Make a graph of that funtion for 0 < A < 4 pi

I don't know what you mean by the "height of the graph".

What? I'm confused ...

My teacher showed us to use the period, critical intervals, max, min and mid points to graph any sine functions

I don't know where you got the "r - r cosA"

Draw a sketch to figure out how much the person rises vertically as the angle A changes. That is what you should get. A=0 is the lowest position.

To find the sine equation for the height of the graph with respect to time, you do not need to find the diameter of the circle. The radius of the Ferris wheel, which is given as 7m, is sufficient for the calculation.

Here's how you can solve this problem step-by-step:

Step 1: Convert the time period to radians. Since the Ferris wheel makes one complete revolution every 16 seconds, you can use the formula:

radians = (2π * t) / T

where t is the time period and T is the period of one revolution. Substituting the given values:

radians = (2π * t) / 16

Step 2: Determine the phase shift. The phase shift is the horizontal translation of the graph. In this case, the person gets onto the Ferris wheel at its lowest point, so the phase shift is 4 seconds (half of the period). This means the graph will start 4 seconds after t = 0.

Step 3: Determine the vertical shift. The bottom of the wheel is 1.5m above the ground, so the vertical shift is 1.5m.

Step 4: Determine the amplitude. The amplitude of the sine function corresponds to half the vertical distance covered by the graph. In this case, the Ferris wheel has a radius of 7m, so the amplitude is 7m.

Step 5: Combine the above information into the sine equation. The general form of a sine function is:

y = A * sin(B(x - C)) + D

where:
- A is the amplitude
- B determines the period
- C determines the phase shift
- D determines the vertical shift

Substituting the values we determined:

y = 7 * sin((2π/16) * (t - 4)) + 1.5

Simplifying:

y = 7 * sin((π/8) * (t - 4)) + 1.5

Finally, in radians, the equation becomes:

h = 7 * sin(π/8 * (t - 4)) + 1.5

So the sine equation for the height (h) of the graph with respect to time (t) is h = 7 * sin(π/8 * (t - 4)) + 1.5. Note that in the given answer, the vertical shift is rounded to the nearest hundredth, resulting in 8.53 instead of 1.5.