Identify the surface with the given vector equation. r(s, t) = s sin 4t, s2, s cos 4t

a. plane
b. hyperbolic paraboloid
c circular paraboloid
d. circular cylinder
e. elliptic cylinder

Circular paraboloid

Step-by-step explanation:

Given ,

Here, these are the respective axes components.

We see that , from the parameterised equation ,

This can also be written as :

This is similar to an equation of a parabola in 1 Dimension.

By fixing the value of z=0,

We get which is equation of a parabola curving towards the positive infinity of y-axis and in the x-y plane.

By fixing the value of x=0,

We get which is equation of a parabola curving towards positive infinity of y-axis and in the y-z plane.

Thus by fixing the values of x and z alternatively ,  we get a CIRCULAR PARABOLOID.

b