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`

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`