Problem 30
Question
Describe geometrically the domain of each of the indicated functions of three variables. \(f(x, y, z)=\frac{\left(144-16 x^{2}-16 y^{2}+9 z^{2}\right)^{3 / 2}}{x y z}\)
Step-by-Step Solution
Verified Answer
The domain is a hyperboloid of two sheets, excluding the planes where \(x = 0\), \(y = 0\), or \(z = 0\).
1Step 1: Identify Allowed Values for the Function's Denominator
The denominator of the function is \(xyz\), which cannot be zero, as division by zero is undefined. Therefore, \(x eq 0\), \(y eq 0\), and \(z eq 0\) must hold for this function to be defined.
2Step 2: Analyze the Expression inside the Square Root
The expression inside the square root is \(144 - 16x^2 - 16y^2 + 9z^2\). For the square root to be real and the function to be defined, we also require \(144 - 16x^2 - 16y^2 + 9z^2 \geq 0\).
3Step 3: Simplify the Inequality
Rearrange the inequality to see what condition \(x\), \(y\), and \(z\) must satisfy: \[16x^2 + 16y^2 - 9z^2 \leq 144\]. Divide through by 144 to simplify:\[\frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1.\]
4Step 4: Interpret the Geometric Region
The inequality \(\frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1\) describes a hyperboloid of two sheets centered at the origin. Since \(x\), \(y\), and \(z\) cannot be zero, the domain does not include the planes where these coordinates are zero.
Key Concepts
Domain of a functionHyperboloidGeometric interpretationInequalities in three variables
Domain of a function
In multivariable calculus, understanding the domain of a function is crucial for knowing where the function is defined. The domain refers to all input values for which a function produces real, admissible outputs. For functions involving three variables, x, y, and z, this involves checking expressions like denominators and radicands (expressions inside square roots).
For the given function, we start by considering the denominator \(xyz\). Division by zero is undefined, so none of the variables x, y, or z can be zero. Therefore, \(x eq 0\), \(y eq 0\), and \(z eq 0\).
Additionally, ensure that the expression under the square root is non-negative. This requires analyzing the inequality \(144 - 16x^2 - 16y^2 + 9z^2 \geq 0\). By solving this inequality, we determine the values that each variable can take, further solidifying our domain.
For the given function, we start by considering the denominator \(xyz\). Division by zero is undefined, so none of the variables x, y, or z can be zero. Therefore, \(x eq 0\), \(y eq 0\), and \(z eq 0\).
Additionally, ensure that the expression under the square root is non-negative. This requires analyzing the inequality \(144 - 16x^2 - 16y^2 + 9z^2 \geq 0\). By solving this inequality, we determine the values that each variable can take, further solidifying our domain.
Hyperboloid
A hyperboloid is a type of quadric surface that can be described geometrically by certain quadratic equations. The hyperboloid of two sheets, which is relevant here, is a three-dimensional surface that resembles two separate bowls touching each other at the narrowest point.
The equation \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1 \) represents a hyperboloid of two sheets. Due to the negative term associated with \(z^2\), it stretches in the z-direction. This specifies that for values of z, the surface consists of two separate pieces or 'sheets.'
The equation \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1 \) represents a hyperboloid of two sheets. Due to the negative term associated with \(z^2\), it stretches in the z-direction. This specifies that for values of z, the surface consists of two separate pieces or 'sheets.'
- In general, hyperboloids can be categorized into one or two sheets depending on their equations.
- They can be rotated around a coordinate axis leading to new forms and equations.
- They appear frequently in physics and engineering as surfaces of revolution.
Geometric interpretation
Geometric interpretation involves visually understanding the properties of mathematical objects through shapes and figures. For the function \( f(x, y, z) \), the geometric interpretation describes where in three-dimensional space this function is defined.
The condition \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1 \) tells us about the region where the hyperboloid of two sheets exists, centered at the origin of the coordinate system.
The condition \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1 \) tells us about the region where the hyperboloid of two sheets exists, centered at the origin of the coordinate system.
- This inequality defines a bounded region in 3D space.
- It is important for determining where the expressions within the function are real and finite.
- By visualizing this region, one can understand how x, y, and z interact to satisfy the domain conditions.
Inequalities in three variables
Inequalities involving three variables are essential to finding regions in space where certain conditions hold.
For this function, the inequality \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1 \) guides which values of \(x\), \(y\), and \(z\) result in the function being properly defined with real numbers.
For this function, the inequality \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} \leq 1 \) guides which values of \(x\), \(y\), and \(z\) result in the function being properly defined with real numbers.
- Each part of the inequality constrains the variables within specific bounds.
- The less-than-or-equal sign indicates a filled region within the boundary defined by the equation \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{16} = 1 \).
- The analysis of these inequalities is crucial for sketching surfaces and understanding spatial relationships.
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