To determine the amplitude of the oscillations after 10 seconds, we need to find the amplitude of the system at that time.

The equation for the amplitude of a damped oscillation is given by:
A(t) = A0 * e^(-ζωnt) * cos(ωdt + φ)

Where:
- A(t) is the amplitude at time t
- A0 is the initial amplitude
- ζ is the damping coefficient
- ωn is the natural frequency of the system
- ωd is the damped frequency of the system

To find the natural frequency (ωn), we can use the formula:
ωn = √(k/m)

Where:
- k is the spring constant
- m is the mass of the system

Since the system is not specified, let's assume the mass (m) and spring constant (k) are both equal to 1 kg.

Therefore, ωn = √(1/1) = 1 rad/s.

We can calculate the damped frequency (ωd) as:
ωd = ωn * √(1 - ζ²)

Substituting the given damping coefficient (ζ = 0.1), we get:
ωd = 1 * √(1 - 0.1²) = 0.9949 rad/s (rounded to four decimal places).

Now we can plug the values into the amplitude equation to find the amplitude after 10 seconds:
A(t) = 0.2 * e^(-0.1 * 1 * 10) * cos(0.9949 * 10 + φ)

Since the system's phase angle (φ) is not mentioned, we'll assume it to be zero.

A(t) = 0.2 * e^(-0.1 * 10) * cos(0.9949 * 10)
≈ 0.2 * e^(-1) * cos(9.949)
≈ 0.2 * 0.3679 * cos(9.949)
≈ 0.0736 * cos(9.949)
≈ 0.0736 * (-0.9898)
≈ -0.0728

Therefore, the amplitude of the oscillations after 10 seconds is approximately -0.0728 meters.

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