Question

At what points does the curve r(t) = ti + (3t − t2)k intersect the paraboloid z = x2 + y2? (if an answer does not exist, enter dne.) (x, y, z) = (smaller t-value) (x, y, z) = (larger t-value)

Answer 1

To find the intersection points of the curve r(t) = ti + (3t − t2)k with the paraboloid z = x2 + y2, we can equate the z values from both equations and solve for t. This leads to a quadratic equation that can be solved to find the values of t at the intersection points.

Step-by-step explanation:

The curve r(t) = ti + (3t − t2)k can be represented in terms of x, y, and z coordinates as x = t, y = 0, and z = 3t − t2. The paraboloid equation z = x2 + y2 can be written as z = t2 since y = 0. Substituting the value of z from the curve equation into the paraboloid equation, we get the equation 3t − t2 = t2. Simplifying this equation gives you the quadratic equation 2t2 − 3t = 0. Solving this quadratic equation will give you the values of t where the curve intersects the paraboloid.

Answer 2

`\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j+z(t)\,\mathbf k=t\,\mathbf i+(3t-t^2)\,\mathbf k`

Along the curve
`\mathbf r(t)`, we have
`y(t)=0`, so the equation of the paraboloid reduces to

`z=x^2\iff3t-t^2=t^2\implies2t^2-3t=t(2t-3)=0\implies t=0\text{ and }t=\frac32`