Question

At what points does the curve r(t) = ti + (4t ? t²)k intersect the paraboloid z = x² + y²? (If an answer does not exist, enter DNE.) Find the smaller t value and the larger t value

Answer 1

To find the points of intersection between the curve and the paraboloid, substitute the equations together and solve for t. The solutions will give the t values at which the curve intersects the paraboloid, and these can be substituted back to find the points of intersection in 3D space.

Step-by-step explanation:

The curve is represented by the equation r(t) = ti + (4t - t²)k. The paraboloid is represented by the equation z = x² + y².

To find the points of intersection, we need to substitute the equations together and solve for t. Substitute x = t, y = 4t - t², and z = t² + (4t - t²)² into the equation for the paraboloid.

After simplifying and rearranging the equation, we get a quadratic equation in t. Use the quadratic formula to solve for t. The solutions will give the t values at which the curve intersects the paraboloid.

Once we have the t values, substitute them back into the equation r(t) = ti + (4t - t²)k to find the corresponding points of intersection in 3D space.