Question

find a function r(t) that describess the curve where the following surfaces intersect. answere are not unique x^2 y^2

Answer 1

Complete Question

find a vector function that represents the curve of intersection of the two surfaces. The cylinder  $x^2+y^2= 36$ an the surface $z=xy$

Answer:

The function is  $r(t) =  6cos(t) \ i + 6sin (t) \ j +  36costsint \ k$

Step-by-step explanation:

From the question we are told that

   The equation of the cylinder is  $x^2+y^2= 36$

   The equation of the surface is  z = xy

Generally the general form of this  function is  

      $r(t) =  x(t)i + y(t)j +  z(t) k$

Generally to confirm the RHS and the LHS of the equation for the cylinder

Let take x (t) =  6cos(t)

and        y(t) = 6sin (t)

So

        $x^2  +  y^2  = [ 6cos(t)]^2 + [6 sin (t)]^2$

=>     $x^2  +  y^2  = 6^2 cos^2t + 6^2 sin ^2t$

=>      $x^2  +  y^2  = 6^2 [cos^2t +  sin ^2t]$    

Generally  $cos^2t +  sin ^2t = 1$

So

        $x^2  +  y^2  = 36$  

So at  x (t) =  6cos(t) and  y(t) = 6sin (t) the RHS is equal to LHS

  So

   $z(t) = x(t) *  y(t)$

    $z(t) = (6 cos(t)) *  (6 sin(t))$

=>     $z(t) =36costsint$

So the function is  

     $r(t) =  6cos(t) i + 6sin (t) j +  36costsint k$