To find the equation of the line tangent to the path of the particle at the given point, we need to find the derivative of the position vector.
The position vector of the particle is given by r(t) = <t^3 + 2t, t^2>.
To find the derivative, we differentiate each component of the vector with respect to t.
Differentiating the x-component of the position vector:
dx/dt = d/dt (t^3 + 2t)
= 3t^2 + 2
Differentiating the y-component of the position vector:
dy/dt = d/dt (t^2)
= 2t
So, the derivative of the position vector is r'(t) = <3t^2 + 2, 2t>.
To find the slope of the tangent line at the point (3, 1), we substitute t = 3 into the derivative:
m = (dy/dt)/(dx/dt)
= (2(3))/(3(3)^2 + 2)
= 6/(27 + 2)
= 6/29
Therefore, the slope of the tangent line at the point (3, 1) is m = 6/29.
To find the equation of the tangent line, we use the point-slope form of a line:
y - y1 = m(x - x1)
Substituting the values of (x1, y1) = (3, 1) and m = 6/29, we get:
y - 1 = (6/29)(x - 3)
Simplifying the equation further, we can write it in the slope-intercept form:
y = (6/29)x - 18/29 + 29/29
y = (6/29)x + 11/29
Therefore, the equation of the tangent line to the path of the particle at the point (3, 1) is y = (6/29)x + 11/29.