To find the rate of change of the particle's distance to the origin, we need to find the rate of change of the particle's distance from the origin, which is the square root of the sum of the squares of the x and y coordinates.

Let's start by finding the equation of the curve in terms of x and y. The given equation y = ln(x^2 + 1) can be rewritten as e^y = x^2 + 1.

Taking the derivative of both sides with respect to x gives e^y * dy/dx = 2x, where dy/dx is the rate of change of y with respect to x.

Dividing both sides by e^y gives dy/dx = 2x / e^y.

Now, we need to find the value of dy/dt, the rate of change of y with respect to time. Since x is increasing at a constant rate of 2 units per second, we have dx/dt = 2.

To find dy/dt, we can use the chain rule:

dy/dt = dy/dx * dx/dt.

Substituting the values we know, we have:

dy/dt = (2x / e^y) * dx/dt.

When x = 3, we can find y by substituting x = 3 into the original equation y = ln(x^2 + 1):

y = ln(3^2 + 1) = ln(10).

Substituting x = 3 and y = ln(10) into the equation for dy/dt, we have:

dy/dt = (2(3) / e^(ln(10))) * 2 = (6 / 10) * 2 = 1.2 units per second.

Therefore, the rate of change of the particle's distance to the origin when x = 3 is 1.2 units per second.

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