SOLUTION: https://gyazo.com/848645a235363970dae8196a58a2bbdc
Algebra.Com
Question 1179653: https://gyazo.com/848645a235363970dae8196a58a2bbdc
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2138) (Show Source): You can put this solution on YOUR website!
Let's break down the problem step-by-step.
**1. Determine the Minimum Height of the Pendulum**
* The rope length is 3 meters.
* The ceiling height is 4 meters.
* The angle of the widest swing is π/3.
* When the pendulum is at its lowest point (vertical), the height from the ceiling is 3 meters.
* When the pendulum swings to its widest point, it forms a triangle with the vertical.
* The vertical component of the rope at the widest swing is 3 * cos(π/3) = 3 * (1/2) = 1.5 meters.
* Therefore, the change in vertical height from the lowest point to the widest point is 3 - 1.5 = 1.5 meters.
* The lowest point of the pendulum is 4 - 3 = 1 meter from the ground.
**2. Determine the Amplitude**
* The amplitude is the change in height from the lowest point to the widest point, which is 1.5 meters.
**3. Determine the Vertical Shift**
* The vertical shift is the height of the pendulum when it is at its lowest point, which is 1 meter from the ground.
* Since the lowest point is 1 meter, the midline of the cosine wave will be 1 + 1.5 = 2.5 meters.
**4. Determine the Period and Angular Frequency**
* The pendulum swings out to its widest position in 2 seconds. This is half of the period.
* Therefore, the full period is 4 seconds.
* The angular frequency (ω) is 2π / period = 2π / 4 = π/2.
**5. Determine the Phase Shift**
* The pendulum is at its lowest point when t = 0.
* A cosine function starts at its maximum value. To model the lowest point at t=0, we need to use a negative cosine.
* There is no horizontal phase shift.
**6. Model the Height Function**
* The general form of a cosine function is: y = A * cos(ωt + φ) + D
* A = Amplitude = 1.5
* ω = Angular frequency = π/2
* φ = Phase shift = 0
* D = Vertical shift = 2.5
* Since we need a negative cosine, we have:
* h(t) = -1.5 * cos(π/2 * t) + 2.5
**Final Answer**
The height of the pendulum from the ground as a function of time is:
h(t) = -1.5cos(πt/2) + 2.5
Answer by ikleyn(53427) (Show Source): You can put this solution on YOUR website!
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https://gyazo.com/848645a235363970dae8196a58a2bbdc
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This solution by @CPhill is irrelevant to the posed problem.
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