Question

At what points does the curve r(t) = ti + (7t - t2)k intersect the paraboloid z = x2 + y2?

Answer 1

To find the points of intersection between the curve r(t) = ti + (7t - t^2)k and the paraboloid z = x^2 + y^2, substitute the values of x, y, and z from the curve equation into the equation of the paraboloid.

Step-by-step explanation:

The given curve is r(t) = ti + (7t - t2)k, and the equation of the paraboloid is z = x2 + y2. To find the points of intersection, we substitute the values of x, y, and z from the curve equation into the equation of the paraboloid:

z = (ti)2 + ((7t - t2)k)2

Simplifying this expression, we get:

z = t2 + (49t2 - 14t3 + t4)

Combining like terms, we obtain:

z = t4 - 14t3 + 50t2

Thus, the points of intersection between the curve and the paraboloid are the values of (t, z) that satisfy this equation.