# 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

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