QuestionName
Date:
THINKING:
16. Make an equation to describe the motion of a point on a wheel that has a centre 5 m off the ground, has a radius of 20 cm , makes a full rotation every 10 seconds and starts at its lowest point.
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17. A rollercoaster at a theme park starts with a vertical drop that leads into two pairs of identical valleys and hills, as shown. Explain at what value the graph starts to exhibit the cosine function. Write the equation of a sinusoidal function to model that part of the graph.
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18. Show that the identity is true for .
Studdy Solution
STEP 1
1. The wheel rotates in a circular motion with a given center and radius.
2. The rollercoaster graph can be modeled using sinusoidal functions.
3. The identity holds for all .
STEP 2
1. Create an equation for the motion of a point on the wheel.
2. Determine where the rollercoaster graph exhibits cosine behavior and model it.
3. Verify the trigonometric identity for a specific angle.
STEP 3
The center of the wheel is 5 meters off the ground, and the radius is 20 cm (0.2 meters).
The vertical position of the point can be described by a sinusoidal function, starting at its lowest point.
STEP 4
Since the wheel makes a full rotation every 10 seconds, the period of the sinusoidal function is 10 seconds.
The amplitude is equal to the radius, 0.2 meters.
The equation for the vertical position of the point is:
STEP 5
Identify the point on the rollercoaster graph where it begins to exhibit cosine behavior.
Assume the graph starts at and the cosine behavior begins at .
STEP 6
The cosine function typically starts at its maximum value. Identify where this occurs.
Write the equation for the sinusoidal function modeling this part of the graph:
Determine , , and based on the graph's amplitude, period, and vertical shift.
STEP 7
Verify the identity for .
Calculate and .
STEP 8
and .
Verify:
The equation for the wheel's motion is .
The cosine behavior starts at with the equation .
The identity is verified for .
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