A pendulum is connected to a rope 3 m long, which is connected to a ceiling 4 m high. The angle between its widest swing and vertical hanging position is pi/3. If the pendulum swings out to its widest position in 2 seconds, model the height of the pendulum from the ground using a cosine function, considering vertical to be t = 0.

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To model the height of the pendulum from the ground using a cosine function, we need to determine the amplitude, frequency, and phase shift.

1. Amplitude: The amplitude of a cosine function represents half the distance between the maximum and minimum values. In this case, the pendulum swings out to its widest position, which would be at a height of 4 m above the ground. Therefore, the amplitude is 4 m.

2. Frequency: The frequency of a cosine function represents the number of cycles completed in one unit of time. In this case, the pendulum swings out to its widest position in 2 seconds. Since one full swing from one side to the other makes one complete cycle, the frequency is 1/2 cycle per second or 0.5 radians per second.

3. Phase Shift: The phase shift of a cosine function represents a horizontal shift of the graph. In this case, the vertical hanging position of the pendulum is considered t = 0. As it swings out to its widest position, the angle formed is pi/3. This means there is a phase shift of pi/3 radians.

Now, we can write the equation for the height of the pendulum from the ground, h(t).

h(t) = A * cos(ωt + φ)

where:
- A is the amplitude,
- ω is the angular frequency (2π times the frequency),
- t is the time,
- φ is the phase shift.

Substituting the values we obtained:

h(t) = 4 * cos(0.5t + π/3)

To model the height of the pendulum using a cosine function, we need to determine the amplitude and period of the function.

First, let's define the variables:
- h(t) represents the height of the pendulum at time t.
- t represents the time in seconds.

Next, let's analyze the given information:
- The pendulum reaches its widest position in 2 seconds. This corresponds to the full period of the cosine function.
- The amplitude of the pendulum's swing is the maximum height it reaches from the vertical hanging position.

To find the amplitude, we need to determine the maximum height the pendulum reaches. From the given information, we know that the pendulum is connected to a rope 3 m long, which is attached to a ceiling 4 m high. The widest swing of the pendulum creates a right triangle with the hypotenuse being the rope and the height being the difference between the rope length and the distance from the ceiling.

Using the Pythagorean theorem, we can determine the height:

Height = sqrt((Rope length)^2 - (Ceiling height)^2)
= sqrt((3)^2 - (4)^2)
= sqrt(9 - 16)
= sqrt(-7)
(Since we are finding the height, we'll ignore the negative square root)

Since the height is an imaginary number, it means that the pendulum does not actually reach that height. Therefore, the amplitude of the cosine function representing the pendulum's swing is 0.

The period of the cosine function can be found using the time it takes for the pendulum to swing out to its widest position. From the given information, the pendulum reaches its widest position in 2 seconds, which corresponds to the full period of the cosine function.

So, the cosine function representing the height of the pendulum can be written as:

h(t) = A * cos(2 * pi * t / P)

where A is the amplitude (0 in this case) and P is the period (2 seconds in this case).

Therefore, the height of the pendulum from the ground (taking vertical as t = 0) can be represented by the equation:

h(t) = 0 * cos(2 * pi * t / 2)
= 0 * cos(pi * t)

Simplifying further, we get:

h(t) = 0

So, the height of the pendulum from the ground is always 0, meaning that it does not go above or below the vertical hanging position.