Calculus III Advanced (Course) (11.6) (Homework)

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Section 11.6 Homework

From Calculus 10e by Larson and Edwards, p. 802. Exercises 1, 2, 3, 4, 5, 6, 10, 16, 32, 44.

Exercises 11.6.1-6 Match the Equation with its Graph

Solution
1. c, 2. e, 3. f, 4. b, 5. d, 6. e.

Exercise 11.6.10 Sketching a Surface in Three Dimensions

\(y^{2}+z=6\)

Solution

Figure 1

Exercise 11.6.16 Sketching a Quadric Surface

Describe and sketch

\( -8x^{2}+18y^{2}+18z^{2}=2 \)

Solution One-sheet Hyperboloid

Figure 2

Exercise 11.6.32 Find the Equation for a Rotated Surface

Curve Equation \(z=3y \)
Coordinate Plane \(yz\)-plane
Axis of Revolution \(y\)-axis

Solution
To find an equation for the surface form by revolving the graph for \(z=3y\) about the \(y\)-axis, solve for \(z\) in \(y\) terms.

\(z=3y=r(y)\)

The equation for the surface is

\( 0x^{2}+z^{2}=[r(y)]^{2} \) Revolved about the \(y\)-axis
\( z^{2}=(3y)^{2} \) Substitute \(3y\) for \(r(y)\)
\( z^{2}=9y^{2} \) Equation for the surface.
Figure 3

Exercise 11.6.44 Find the Equation for a Surface

The set for all points equidistant from the point (0,0,4) and the \(xy\)-plane.
Solution
Find the distance from the point \(x=y\) and \(z=0\) to the point (0,0,4).

\( d=\sqrt{x^{2}+y^{2}+(z-4)^{2}} \)

The length is the line from \((x,y,z)\) to the \(xy\)-plane is \( |z| \) so

\( |z| \) \(= \sqrt{x^{2}+y^{2}+(z-4)^{2}} \)
\(= x^{2}+y^{2}+z^{2}-8z+16 \)
\( 0 \) \(= x^{2}+y^{2}-8z+16 \)
Figure 4

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Parent Article: Calculus III Advanced (Course)