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

## Contents

# 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**

## Exercise 11.6.16 Sketching a Quadric Surface

Describe and sketch

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

**Solution** One-sheet Hyperboloid

## 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.

## 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

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

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

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

# Internal Links

*Parent Article:* Calculus III Advanced (Course)