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Chapter 10: Problem 26

Sketch the appropriate traces, and then sketch and identify the surface. $$z=9-x^{2}-y^{2}$$

Short Answer

Expert verified
The given equation represents the upper half of a cone with vertex at (0, 0, 9) opening downwards.

Step by step solution

01

Identify the Form of the Equation

Examining the equation, it can be rearranged into the form \(x^{2} + y^{2} = z - 9\), this is the equation of a circle in 3-dimensional space. That tells us that for each value of \(z\), we have a circle. When \(z > 9\), the circle is imaginary which we can't plot. Therefore, the solution has a range \(z \leq 9\). This analysis will define the range of the surface.
02

Plot the Traces

There are two types of traces that we can plot:Xy-trace: When \(z = 0\), the equation becomes \(x^{2} + y^{2} = 9\), which is a circle of radius 3. Xz-trace: When \(y = 0\), the equation becomes \(-x^{2} + z = 9\), or \(z = x^{2} + 9\). This is a parabola opening upwards. These traces help define the shape and orientation of the surface.
03

Sketch the Surface

Using the traces, now sketch the surface. You will see that it is the upper part of a cone which has its vertex at the point (0, 0, 9) and opens downwards.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Sphere
The equation of a sphere in three-dimensional space has a standard form: \[ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \] where
  • \((h, k, l)\) are the coordinates of the center of the sphere
  • \(r\) is the radius of the sphere
For instance, the equation of a sphere centered at the origin with a radius of 3 would be \(x^2 + y^2 + z^2 = 9\).
This form shows how each point \((x, y, z)\) on the sphere is equidistant from the center.
Changing the center or the radius allows you to identify any sphere in the three-dimensional space.
Parabolic Cylinder
A parabolic cylinder is an infinite surface that extends in a direction orthogonal to a parabola.
The standard form in 3D is \[ z = x^2 + c \] where \(c\) can be any constant.
  • It opens in the \(z\)-direction when \(y\) is kept constant.
  • The cross-sections parallel to the \(yz\)-plane form a set of identical parabolas.
These parabolas help in sketching the parabolic cylinder's shape, essentially stretching the parabola infinitely along the \(z\)-axis.
It鈥檚 important to remember that despite its name, a cylinder doesn鈥檛 have to be circular.
Being aware of how it extends through space helps visualize complex surfaces.
Surface Sketching
Surface sketching involves understanding and visualizing different traces or cross-sections. For the given exercise,
  • The \(xy\)-trace is \(x^2 + y^2 = z\)
  • The \(xz\)-trace is \(-x^2 + z = 9\)
Here's how to approach it:
  • Start by sketching each trace. For example, a circle and a parabola.
  • Analyze changes in these shapes as you move along different axes.
These traces combine to display the entire surface, making it easier to identify complex surfaces like the one in the exercise鈥攁 paraboloid.
Once you understand the individual parts, you can sketch the full 3D surface effectively.

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Most popular questions from this chapter

Find the distance between the given objects. Show that the distance between planes \(a x+b y+c z=d_{1}\) and \(a x+b y+c z=d_{2}\) is given by \(\frac{\left|d_{2}-d_{1}\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

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