To find the parametric equations for the curve of intersection between the surfaces:

1. Cone equation: $z = \sqrt{x^2 + y^2}$
2. Plane equation: $z = 4 + y$

Step 1: Set the $z$-coordinates equal

Since both surfaces share the same $z$-value at their intersection, we can set the equations for $z$ equal to each other:

$\sqrt{x^2 + y^2} = 4 + y$

Step 2: Square both sides

To eliminate the square root, square both sides of the equation:

$x^2 + y^2 = (4 + y)^2$

Expanding the right-hand side:

$x^2 + y^2 = 16 + 8y + y^2$

Step 3: Simplify the equation

Subtract $y^2$ from both sides to eliminate $y^2$:

$x^2 = 16 + 8y$

This simplifies to:

$x^2 = 8(y + 2)$

Step 4: Solve for $x$

We can now solve for $x$:

$x = \pm \sqrt{8(y + 2)}$

For simplicity, choose the positive root:

$x = \sqrt{8(y + 2)}$

Step 5: Parametrize $y$

We can let $y$ be a parameter. Let's define $y = t$, so the parametric equations will be in terms of $t$. Thus:

$x = \sqrt{8(t + 2)}$ $y = t$

Step 6: Parametrize $z$

Now, using the plane equation $z = 4 + y$, we substitute $y = t$:

$z = 4 + t$

Final Parametric Equations

The parametric equations for the curve of intersection are:

$x(t) = \sqrt{8(t + 2)}$ $y(t) = t$ $z(t) = 4 + t$

Let me know if you would like further details or clarifications!

Here are 5 related questions to further explore the topic:

1. How do the parametric equations describe the curve geometrically?
2. What happens to the curve if the plane equation is modified to $z = 5 + y$?
3. Can you parametrize the curve using polar coordinates?
4. How would the curve change if the cone equation was $z = 2\sqrt{x^2 + y^2}$?
5. What is the geometric significance of the parameter $t$ in this case?

Tip: When squaring both sides of an equation, always check for extraneous solutions introduced by the squaring process!