The problem asks to identify and sketch the graph of equations in three-dimensional space. We will solve problems 16, 17, 18, 19 and 20.

1. Problem Description

The problem asks to identify and sketch the graph of equations in three-dimensional space. We will solve problems 16, 17, 18, 19 and

2

0.

2. Solution Steps

Problem 16: $9x^2 + 25y^2 + 9z^2 = 225$

Divide both sides by 225:

$\frac{9x^2}{225} + \frac{25y^2}{225} + \frac{9z^2}{225} = 1$

$\frac{x^2}{25} + \frac{y^2}{9} + \frac{z^2}{25} = 1$

$\frac{x^2}{5^2} + \frac{y^2}{3^2} + \frac{z^2}{5^2} = 1$

This is an ellipsoid. Since the coefficients of $x^2$ and $z^2$ are the same after normalization, the ellipsoid is symmetric around the y-axis.

Problem 17: $5x + 8y - 2z = 10$

This is a linear equation in three variables, which represents a plane.

Problem 18: $y = \cos x$

This equation does not involve $z$. Thus, the graph is a cosine curve in the $xy$-plane, extended infinitely along the $z$-axis. This forms a cosine cylinder.

Problem 19: $z = \sqrt{16 - x^2 - y^2}$

Square both sides:

$z^2 = 16 - x^2 - y^2$

$x^2 + y^2 + z^2 = 16$

$x^2 + y^2 + z^2 = 4^2$

This is a sphere centered at the origin with radius

4. Since $z = \sqrt{16 - x^2 - y^2}$, $z$ must be non-negative, $z \ge 0$. Therefore, the graph is the upper hemisphere.

Problem 20: $z = \sqrt{x^2 + y^2} + 1$

Square both sides after subtracting

1. $z-1 = \sqrt{x^2 + y^2}$

$(z-1)^2 = x^2 + y^2$

This represents a cone with vertex at $(0, 0, 1)$ opening along the positive z-axis, shifted up by

1. Since $z=\sqrt{x^2+y^2}+1$, we require $z\ge 1$

3. Final Answer

Problem 16: Ellipsoid.

Problem 17: Plane.

Problem 18: Cosine Cylinder.

Problem 19: Upper Hemisphere.

Problem 20: Cone with vertex at $(0,0,1)$ opening along the positive $z$-axis, $z \ge 1$.