Problem 33
Question
Find the domain of the following functions. $$ z=\sqrt{100-4 x^{2}-25 y^{2}} $$
Step-by-Step Solution
Verified Answer
The domain is \( \frac{x^2}{25} + \frac{y^2}{4} \leq 1 \).
1Step 1: Understand the Function
The function given is \( z = \sqrt{100 - 4x^2 - 25y^2} \). To find its domain, we need to determine all the possible values of \( x \) and \( y \) that make the expression under the square root function non-negative as square root is defined only for non-negative numbers.
2Step 2: Setup the Inequality
For the square root to be defined, the expression under it must be greater than or equal to zero. Therefore, we need to solve the inequality \( 100 - 4x^2 - 25y^2 \geq 0 \).
3Step 3: Rearrange the Inequality
Rearrange the inequality \( 100 - 4x^2 - 25y^2 \geq 0 \) to: \( 4x^2 + 25y^2 \leq 100 \). This can be further simplified to \( \frac{x^2}{25} + \frac{y^2}{4} \leq 1 \).
4Step 4: Interpret the Inequality
The inequality \( \frac{x^2}{25} + \frac{y^2}{4} \leq 1 \) represents the equation of an ellipse centered at \((0,0)\). The semi-major axis is 5 units along the x-axis, while the semi-minor axis is 2 units along the y-axis.
5Step 5: Define the Domain
The domain of the function comprises all the pairs \((x, y)\) that satisfy \( \frac{x^2}{25} + \frac{y^2}{4} \leq 1 \). In simpler terms, it consists of all the points contained within and on the ellipse defined by \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \).
Key Concepts
InequalitiesEllipsesSquare Root Function
Inequalities
In mathematics, inequalities are expressions that involve relational symbols such as \(\geq \), \(\leq \), \(>\), and \(<\). These symbols indicate that one side of an inequality is either larger, smaller, or equal to some extent in comparison to the other side. In the context of function domains, inequalities are crucial as they can help determine which values of variables keep a function valid.
For our function \(z = \sqrt{100 - 4x^2 - 25y^2}\), we deal with the inequality \(100 - 4x^2 - 25y^2 \geq 0\). This form ensures that the expression inside the square root is non-negative, allowing the function to maintain real number outputs. This requirement, due to the square root operation, leads us to define the inequality \(4x^2 + 25y^2 \leq 100\) for the domain.
Use of inequalities simplifies translating the conditions that must be met for variables within functions. This helps in visualizing limits on input values, which is essential when sketching functions or determining their physical constraints. Inequalities thus serve both a theoretical and practical purpose in exploring how functions behave across different domains.
For our function \(z = \sqrt{100 - 4x^2 - 25y^2}\), we deal with the inequality \(100 - 4x^2 - 25y^2 \geq 0\). This form ensures that the expression inside the square root is non-negative, allowing the function to maintain real number outputs. This requirement, due to the square root operation, leads us to define the inequality \(4x^2 + 25y^2 \leq 100\) for the domain.
Use of inequalities simplifies translating the conditions that must be met for variables within functions. This helps in visualizing limits on input values, which is essential when sketching functions or determining their physical constraints. Inequalities thus serve both a theoretical and practical purpose in exploring how functions behave across different domains.
Ellipses
An ellipse is a geometric shape that can be thought of as a stretched circle. A standard form for an ellipse equation centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, the constants \(a\) and \(b\) determine the length of the axes. The larger of these values dictates the ellipse's semi-major axis, while the smaller controls the semi-minor axis.
In our function's inequality, \(\frac{x^2}{25} + \frac{y^2}{4} \leq 1\), we essentially describe the space contained within and on an ellipse. The terms \(25\) and \(4\) represent \(a^2\) and \(b^2\) respectively, pointing to the semi-major axis along the x-axis being 5 units long \(\sqrt{25} = 5\), and the semi-minor axis along the y-axis being 2 units \(\sqrt{4} = 2\).
Understanding that this inequality covers an ellipse tells us about the region where \(x\) and \(y\) are allowed to roam while still keeping the function defined. Since this ellipse is centered at the origin \(\left(0, 0\right)\), it highlights a symmetrical area around \(x\) and \(y\) axes where our function can take real values.
In our function's inequality, \(\frac{x^2}{25} + \frac{y^2}{4} \leq 1\), we essentially describe the space contained within and on an ellipse. The terms \(25\) and \(4\) represent \(a^2\) and \(b^2\) respectively, pointing to the semi-major axis along the x-axis being 5 units long \(\sqrt{25} = 5\), and the semi-minor axis along the y-axis being 2 units \(\sqrt{4} = 2\).
Understanding that this inequality covers an ellipse tells us about the region where \(x\) and \(y\) are allowed to roam while still keeping the function defined. Since this ellipse is centered at the origin \(\left(0, 0\right)\), it highlights a symmetrical area around \(x\) and \(y\) axes where our function can take real values.
Square Root Function
The square root function, denoted as \(\sqrt{x}\), is a basic mathematical function that outputs the number which, when squared, equals \(x\). This function is only defined for non-negative inputs since the square root of negative numbers shifts the result into the realm of imaginary numbers, which we generally don't deal with in real number functions.
In the given exercise, \(z = \sqrt{100 - 4x^2 - 25y^2}\), the square root function places a key restriction: the expression inside the square root, \(100 - 4x^2 - 25y^2\), must not dip below zero. This constraint was why we tackled the inequality \(4x^2 + 25y^2 \leq 100\), ensuring that our function returns only real numbers.
The history of the square root function goes back to ancient civilizations, illustrating its fundamental importance and utility in solving practical mathematical problems. Today, the ability to compute square roots is embedded in virtually every field of science and engineering, underlining its broad significance.
In the given exercise, \(z = \sqrt{100 - 4x^2 - 25y^2}\), the square root function places a key restriction: the expression inside the square root, \(100 - 4x^2 - 25y^2\), must not dip below zero. This constraint was why we tackled the inequality \(4x^2 + 25y^2 \leq 100\), ensuring that our function returns only real numbers.
The history of the square root function goes back to ancient civilizations, illustrating its fundamental importance and utility in solving practical mathematical problems. Today, the ability to compute square roots is embedded in virtually every field of science and engineering, underlining its broad significance.
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