Question

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

Answer 1

Answer:

Circular paraboloid

Step-by-step explanation:

Given ,

$r(s,t)=ssin4t,s^2,scos4t$

Here, these are the respective $x,y,z$ axes components.

• Component along x axis $r_i : ssin4t$

• Component along y axis $r_j : s^2$

• Component along z axis $r_k : scos4t$

We see that , from the parameterised equation , $r_i^2+r_k^2=s^2sin^24t+s^2cos^24t\\r_i^2+r_k^2=s^2\\r_i^2+r_k^2=r_j$

This can also be written as :

$x^2+z^2=y$

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

By fixing the value of z=0,

We get $y=x^2$ 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 $y=z^2$ 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.