We are asked to describe geometrically the domain of the following functions of three variables: 27. $f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16}$ 28. $f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9}$ 29. $f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2}$

1. Problem Description

We are asked to describe geometrically the domain of the following functions of three variables:

2

7. $f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16}$

2

8. $f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9}$

2

9. $f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2}$

2. Solution Steps

2

7. For the function $f(x, y, z) = \sqrt{x^2 + y^2 + z^2 - 16}$, the domain is the set of all points $(x, y, z)$ such that the expression inside the square root is non-negative. Thus, we have

$x^2 + y^2 + z^2 - 16 \ge 0$

$x^2 + y^2 + z^2 \ge 16$

This represents all points on or outside the sphere with center $(0, 0, 0)$ and radius

4.

2

8. For the function $f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9}$, the domain is the set of all points $(x, y, z)$ such that the expression inside the square root is non-negative. Thus, we have

$x^2 + y^2 - z^2 - 9 \ge 0$

$x^2 + y^2 - z^2 \ge 9$

$x^2 + y^2 \ge z^2 + 9$

This represents all points on or outside the hyperboloid of two sheets with equation $x^2 + y^2 - z^2 = 9$.

2

9. For the function $f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2}$, the domain is the set of all points $(x, y, z)$ such that the expression inside the square root is non-negative. Thus, we have

$144 - 16x^2 - 9y^2 - 144z^2 \ge 0$

$16x^2 + 9y^2 + 144z^2 \le 144$

Dividing by 144, we get

$\frac{16x^2}{144} + \frac{9y^2}{144} + \frac{144z^2}{144} \le 1$

$\frac{x^2}{9} + \frac{y^2}{16} + \frac{z^2}{1} \le 1$

This represents all points on or inside the ellipsoid $\frac{x^2}{9} + \frac{y^2}{16} + \frac{z^2}{1} = 1$ with semi-axes 3, 4, and 1 along the x, y, and z axes, respectively.

3. Final Answer

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7. The domain is all points on or outside the sphere with center $(0, 0, 0)$ and radius

4.

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8. The domain is all points on or outside the hyperboloid of two sheets with equation $x^2 + y^2 - z^2 = 9$.

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9. The domain is all points on or inside the ellipsoid $\frac{x^2}{9} + \frac{y^2}{16} + \frac{z^2}{1} = 1$.