Final answer:
To find the intersection points of the curve with the paraboloid, the expressions for x, y, and z are substituted from the curve into the paraboloid's equation and then solved for t, yielding two intersection points at t-values 0 and 1.5.
Step-by-step explanation:
To find the intersection points of the curve r(t) = ti + (3t - t2)k with the paraboloid z = x2 + y2, we need to substitute the expressions for x, y, and z from the curve into the equation of the paraboloid and solve for t.
- x is given by the coefficient of i in r(t), which is t.
- y is not present in r(t), which implies that y = 0.
- z is given by the coefficient of k in r(t), which is (3t - t2).
Substituting into the paraboloid's equation, we get 3t - t2 = t2. Simplifying, 3t = 2t2 or t(t - 1.5) = 0, giving us the solutions t = 0 and t = 1.5. Thus, the curve intersects the paraboloid at these two t-values.