Problem 29
Question
Describe geometrically the domain of each of the indicated functions of three variables. \(f(x, y, z)=\sqrt{144-16 x^{2}-9 y^{2}-144 z^{2}}\)
Step-by-Step Solution
Verified Answer
The domain is the interior and surface of an ellipsoid centered at the origin.
1Step 1: Recognize the function type
The function provided is \( f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2} \). This is a square root function, and to ensure the values under the square root are defined (non-negative), we need the expression inside the square root to be greater than or equal to zero: \( 144 - 16x^2 - 9y^2 - 144z^2 \geq 0 \).
2Step 2: Set up the inequality
Using the inequality from Step 1, rearrange it: \( 16x^2 + 9y^2 + 144z^2 \leq 144 \). This represents the set of points \((x, y, z)\) inside an ellipsoid.
3Step 3: Simplify the inequality
Divide the inequality by 144 to simplify it further: \( \frac{x^2}{9} + \frac{y^2}{16} + z^2 \leq 1 \). This is the standard equation of an ellipsoid centered at the origin with semi-axis lengths of 3, 4, and 1.
4Step 4: Geometric description of the domain
The domain of the function geometrically is the region inside the ellipsoid \( \frac{x^2}{9} + \frac{y^2}{16} + z^2 \leq 1 \). This implies that points \((x, y, z)\) must lie inside or on the surface of this ellipsoid.
Key Concepts
Domain of a FunctionEllipsoidSquare Root Function
Domain of a Function
In multivariable calculus, the concept of the domain of a function is vital. The domain essentially tells us where a function is defined — that is, the set of all possible input values. For functions of several variables, like our function of three variables, determining the domain can help in understanding the geometric and spatial behavior of the function.
To find the domain of the function given by \( f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2} \), we must ensure that the values under the square root are non-negative. This is crucial because square roots of negative numbers aren't real, which would make our function undefined.
By rearranging the inequality \( 144 - 16x^2 - 9y^2 - 144z^2 \geq 0 \), we identify where the function lives. Doing so leads to the inequality \( \frac{x^2}{9} + \frac{y^2}{16} + z^2 \leq 1 \). These are the points where the function is "happy" and defined. Thus, the domain is the entirety of space contained by this inequality, including its boundary.
To find the domain of the function given by \( f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2} \), we must ensure that the values under the square root are non-negative. This is crucial because square roots of negative numbers aren't real, which would make our function undefined.
By rearranging the inequality \( 144 - 16x^2 - 9y^2 - 144z^2 \geq 0 \), we identify where the function lives. Doing so leads to the inequality \( \frac{x^2}{9} + \frac{y^2}{16} + z^2 \leq 1 \). These are the points where the function is "happy" and defined. Thus, the domain is the entirety of space contained by this inequality, including its boundary.
- The domain will always correspond to some tangible region in space when graphed.
- It is necessary to consider all constraints imposed by the function definition, such as square roots and division by zero.
- Here, the domain is an ellipsoid, which is a three-dimensional analogue of an ellipse.
Ellipsoid
An ellipsoid is a rounded, three-dimensional geometric figure, resembling an elongated or flattened sphere. When expressed mathematically, an ellipsoid's standard form can help elucidate the territory where a particular function is defined.
The function \( f(x, y, z)\) is said to be inside an ellipsoid if \(\frac{x^2}{9} + \frac{y^2}{16} + z^2 \leq 1\). This equation represents an ellipsoid centered at the origin of the coordinate system. The numbers in the denominators (9, 16, and 1 for \(x, y, z\), respectively) correspond to the squares of the lengths of the semi-axes.
The semi-axes give a clue about the ellipsoid's shape and size:
The function \( f(x, y, z)\) is said to be inside an ellipsoid if \(\frac{x^2}{9} + \frac{y^2}{16} + z^2 \leq 1\). This equation represents an ellipsoid centered at the origin of the coordinate system. The numbers in the denominators (9, 16, and 1 for \(x, y, z\), respectively) correspond to the squares of the lengths of the semi-axes.
The semi-axes give a clue about the ellipsoid's shape and size:
- The length of the semi-axis along the x-direction is 3 (since \(\sqrt{9} = 3\)).
- The length of the semi-axis along the y-direction is 4 (since \(\sqrt{16} = 4\)).
- The length of the semi-axis along the z-direction is 1.
Square Root Function
The square root function is a fundamental mathematical operation. It involves finding a number that, when multiplied by itself, yields the given number. In calculus, the square root function frequently appears, particularly in problems involving distances and rates of change.
For the given multivariable function \(f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2}\), the square root notation denotes an operation on three variables. The constraints from the square root inform us about the valid region where the function makes sense - where the value under the square root remains non-negative.
In real-world applications and in geometry, square roots help measure the true distance and magnitude of vectors or geometric shapes:
For the given multivariable function \(f(x, y, z) = \sqrt{144 - 16x^2 - 9y^2 - 144z^2}\), the square root notation denotes an operation on three variables. The constraints from the square root inform us about the valid region where the function makes sense - where the value under the square root remains non-negative.
In real-world applications and in geometry, square roots help measure the true distance and magnitude of vectors or geometric shapes:
- The square root function ensures outputs are non-negative, which aligns well with physical quantities like distances.
- It reflects a pivotal role in determining geometric boundaries, like the contours of an ellipsoid.
- Square roots often simplify complex physical models into interpretable mathematical forms.
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