Problem 60 - Calculus III

Question

Find a vector function that describes the following curves. Intersection of the cone \(z=\sqrt{x^{2}+y^{2}}\) and plane \(z=y-4\)

Step-by-Step Solution

Answer

The vector function is \(\mathbf{r}(t) = (\pm\sqrt{16 - 8t}, t, t - 4)\) for \(4 \leq t \leq 8\).

Step 1: Set Equations Equal

To find the intersection, equate the two given surfaces: the cone equation \(z = \sqrt{x^2 + y^2}\) and the plane equation \(z = y - 4\). Thus, we have \(\sqrt{x^2 + y^2} = y - 4\).

Step 2: Solve for y

Square both sides to eliminate the square root: \(x^2 + y^2 = (y - 4)^2\). Expanding the right side gives \(y^2 - 8y + 16\). So the equation becomes \(x^2 + y^2 = y^2 - 8y + 16\).

Step 3: Simplify the Equation

Cancel \(y^2\) from both sides and rearrange to find \(x^2 = 16 - 8y\). Thus, we have \(x^2 = 16 - 8y\).

Step 4: Express y in terms of a parameter

Set \(y = t\), which implies \(x^2 = 16 - 8t\). Then \(x = \pm\sqrt{16 - 8t}\).

Step 5: Substitute Parametric y into Plane Equation for z

Substitute \(y = t\) into \(z = y - 4\) to get \(z = t - 4\).

Step 6: Combine into Vector Function

Assemble the parametric equations into a vector function: \(\mathbf{r}(t) = (\pm\sqrt{16 - 8t}, t, t - 4)\), for \(4 \leq t \leq 8\) (determined by \(16 - 8t \geq 0\)).

Step 7: Finalize Vector Function and Domain

The vector function describing the intersection is \(\mathbf{r}(t) = (\pm\sqrt{16 - 8t}, t, t - 4)\), with \(4 \leq t \leq 8\) as its domain.

Key Concepts

Intersection of Surfaces

When you are asked to find the intersection of two surfaces, it's like looking for the common points shared by two different shapes in space. In mathematics, surfaces can be described with equations.
For example, a cone might be defined with a formula showing how tall it is at any given point, while a plane could have an equation showing how high the plane is at every point.

• The cone equation might be something like \( z=\sqrt{x^{2}+y^{2}} \). This indicates that as you move away from the center, the height (z) increases based on the distance from the x and y coordinates.
• The plane might be given as \( z=y-4 \), describing a flat surface that shifts in height depending on the y position.

To find their intersection, we set the two equations equal to each other. This gives us a new equation representing just the points where the cone and plane meet.

Parametric Equations

Parametric equations are a great way of describing curves. Instead of expressing y directly in terms of x (or vice versa), you introduce a third variable, often called a parameter (like t).
This parameter helps to describe the position on the curve with one equation for x, one for y, and another for z, if working in three dimensions.
This method is really helpful when dealing with intersections of surfaces.

• For instance, after setting the two surface equations equal, you might solve for y in terms of this parameter. Like setting \( y = t \), leading us to other expressions like \( x = \pm\sqrt{16 - 8t} \).
• Then, replacing y in the plane's equation to find z gives \( z = t - 4 \).

Finally, these three are combined into a vector function, which is a vector consisting of (x, y, z) each expressed in terms of t. This beautifully wraps up the information about a curve into one neat package.

Cone and Plane Intersection

When a cone intersects with a plane, the result is often an interesting curve. This is a common scenario in mathematics called a conic section, sharing the class of curves like circles, ellipses, parabolas, and hyperbolas.
In this specific example, the intersection is examined by looking at both the cone's and the plane's equations.

• The cone has the equation \( z = \sqrt{x^2 + y^2} \), which tells us the cone's surface is where the height is influenced by the distance from the origin in the xy-plane.
• The plane is described as \( z = y - 4 \), indicating it slopes upwards as y increases.

By setting these two equations equal (\( \sqrt{x^2 + y^2} = y - 4 \)), and transforming it through algebra, you can derive a relationship between x and y.
In our example, it produced \( x^2 = 16 - 8y \). Using a parameter like \( y = t \), you can then represent this intersection as a parametric curve, with boundaries defined by \( 4 \leq t \leq 8 \).
This restricts the curve to the section of the intersection that logically fits both surfaces.