To create a model for the height of the heart over time, we first need to define some variables and use the information given.
Let's choose:
- Diameter of the wheel = 26 inches
- Frequency of the wheel = 70 revolutions per minute
Step A: Determine the parameters for the equation
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Radius of the Wheel: \[ r = \frac{\text{diameter}}{2} = \frac{26}{2} = 13 \text{ inches} \]
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Circumference of the Wheel: The height of the heart follows a sinusoidal pattern. The vertical movement can be modeled as a cosine function: \[ \text{Circumference} = 2 \pi r = 2 \pi \times 13 \approx 81.68 \text{ inches} \]
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Height Function: The height of the heart (h) above the ground as a function of time (x) in minutes can be modeled as: \[ h(x) = r \cdot \cos(2 \pi f x) + r \] where \( f \) is the frequency in revolutions per minute, and the cosine function is adjusted to start at the maximum height. Substituting in our values: \[ h(x) = 13 \cos(2 \pi \cdot 70 x) + 13 \]
Step B: Evaluate h at 30 seconds and 5 minutes
Note: We need to convert time into minutes since our function is defined in minutes.
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30 seconds into minutes: \[ 30 \text{ seconds} = \frac{30}{60} = 0.5 \text{ minutes} \] Now, calculate \( h(0.5) \): \[ h(0.5) = 13 \cos(2 \pi \cdot 70 \cdot 0.5) + 13 \] \[ h(0.5) = 13 \cos(70 \pi) + 13 \] Since \( \cos(70 \pi) = -1 \): \[ h(0.5) = 13 \cdot (-1) + 13 = -13 + 13 = 0 \text{ inches} \]
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5 minutes into the ride: \[ h(5) = 13 \cos(2 \pi \cdot 70 \cdot 5) + 13 \] \[ h(5) = 13 \cos(700 \pi) + 13 \] Since \( \cos(700 \pi) = 1 \): \[ h(5) = 13 \cdot 1 + 13 = 13 + 13 = 26 \text{ inches} \]
Final Answers:
A. The height of the heart exactly 30 seconds into the ride is 0 inches.
B. The height of the heart exactly 5 minutes into the ride is 26 inches.