Final answer:
To find the points of intersection between the curve r(t) = ti + (4t + t²)k and the paraboloid z = x² + y², substitute the values of x, y, and z from the curve equation into the equation of the paraboloid.
Step-by-step explanation:
To find the points of intersection between the curve r(t) = ti + (4t + t²)k and the paraboloid z = x² + y², we need to substitute the values of x, y, and z from the curve equation into the equation of the paraboloid.
Given that r(t) = ti + (4t + t²)k, we can find x and y by equating the coefficients of the i and j unit vectors, respectively. So, x = t and y = 4t + t².
Substituting these values into the equation of the paraboloid, we get z = (t)² + (4t + t²)². Simplifying this equation will give us the points of intersection.
To determine at what points the curve r(t) = ti + (4t + t²)k intersects the paraboloid z = x² + y², we need to find the values of t for which the z-coordinate of the curve matches the z-coordinate of the paraboloid.
The curve given has no j (y) component, which implies that y = 0. The x-coordinate of the curve is t, and since y is zero, the z-coordinate of the curve is given as 4t + t².
Now, we will equate the z-coordinate of the curve to the z-value in the paraboloid equation:
4t + t² = t² + 0²
4t + t² = t²
4t = 0
From the last equation, it is clear that t = 0. Thus, the curve intersects the paraboloid at the origin, which is the only intersection point.